A determinant of second order is made with the elements 0 and 1. The number of determinants with non - negative values is:
- 3
- 10
- 11
- 13
Question types
DETERMINANTS question types
698 questions across 9 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
698
Questions
9
Question groups
5
Question types
01
M.C.Q (1 Marks)
170 Q→02Assertion (A) & Reason (B) MCQ
17 Q→031 Marks Question
27 Q→042 Marks Questions
110 Q→053 Marks Question
73 Q→065 Marks Questions
270 Q→07Case study (4 Marks)
10 Q→08Fill In The Blanks[1 Marks ]
10 Q→09True False[1 Marks ]
11 Q→Sample Questions
DETERMINANTS questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
View full solution →
If A is a singular matrix, then A (adj A) is a
- scalar matrix
- zero matrix
- identity matrix
- orthogonal matrix
Evaluate $\begin{bmatrix}4&8&12\\6&12&18\\7&14&21\end{bmatrix}$ is:
View full solution →- 168
- -1
- -168
- 0
Evaluate $\begin{bmatrix}\text{x}^2&\text{x}^3&\text{x}^4\\\text{x}&\text{y}&\text{z}\\\text{x}^2&\text{x}^3&\text{x}^4\end{bmatrix}$ is:
- ✓$0$
- B$1$
- C$xyz$
- D$x^2 yz^3$
Answer: A.
View full solution → Evaluate $\begin{bmatrix}1&0&1\\0&0&1\\1&0&1\end{bmatrix}$ is:
View full solution →- 2
- 0
- 1
- -1
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: For a matrix $\begin{bmatrix}2&-1\\-3&4\end{bmatrix},$ A. adj $\text{A}=\begin{bmatrix}4&0\\0&4\end{bmatrix}.$
Reason: For a square matrix A, $\text{A}(\text{adj}\text{A})=(\text{adj}\text{A})\text{A}=\mid\text{A}\mid.$
View full solution →Assertion: For a matrix $\begin{bmatrix}2&-1\\-3&4\end{bmatrix},$ A. adj $\text{A}=\begin{bmatrix}4&0\\0&4\end{bmatrix}.$
Reason: For a square matrix A, $\text{A}(\text{adj}\text{A})=(\text{adj}\text{A})\text{A}=\mid\text{A}\mid.$
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If $\text{A}=\begin{bmatrix}1&0&1\\0&1&2\\0&0&4\end{bmatrix}$ then $\mid3\text{A}\mid=9\mid\text{A}\mid.$
Reason: If A is a square matrix of order n then $\mid\text{k}\text{A}\mid=\text{k}^{\text{n}}\mid\text{A}\mid.$
View full solution →Assertion: If $\text{A}=\begin{bmatrix}1&0&1\\0&1&2\\0&0&4\end{bmatrix}$ then $\mid3\text{A}\mid=9\mid\text{A}\mid.$
Reason: If A is a square matrix of order n then $\mid\text{k}\text{A}\mid=\text{k}^{\text{n}}\mid\text{A}\mid.$
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: The value of x for which $\begin{vmatrix}\text{x}&2\\18&\text{x}\end{vmatrix}=\begin{vmatrix}6&2\\18&6\end{vmatrix}$ is $\pm\ 6.$
Reason: The determinant of a matrix A order 2x2, $\text{A}\begin{bmatrix}\text{a}&\text{b}\\\text{c}&\text{d}\end{bmatrix}$ is = ab - dc.
View full solution →Assertion: The value of x for which $\begin{vmatrix}\text{x}&2\\18&\text{x}\end{vmatrix}=\begin{vmatrix}6&2\\18&6\end{vmatrix}$ is $\pm\ 6.$
Reason: The determinant of a matrix A order 2x2, $\text{A}\begin{bmatrix}\text{a}&\text{b}\\\text{c}&\text{d}\end{bmatrix}$ is = ab - dc.
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: For two matrices A and B of order 3, $\mid\text{A}\mid=2\mid\text{B}\mid=-3$ then if $\mid2\text{AB}\mid$ is -48.
Reason: For a square matrix A, $\text{A}(\text{adj}\ \text{A})=(\text{adj}\ \text{A})\text{A}=\mid\text{A}\mid.$
View full solution →Assertion: For two matrices A and B of order 3, $\mid\text{A}\mid=2\mid\text{B}\mid=-3$ then if $\mid2\text{AB}\mid$ is -48.
Reason: For a square matrix A, $\text{A}(\text{adj}\ \text{A})=(\text{adj}\ \text{A})\text{A}=\mid\text{A}\mid.$
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: The value of x for which $\begin{vmatrix}3&\text{x}\\\text{x}&1\end{vmatrix}=\begin{vmatrix}3&2\\4&1\end{vmatrix}$ is $\pm2\sqrt{2}.$
Reason: The determinant of a matrix A order 2x2, $\text{A}\begin{bmatrix}\text{a}&\text{b}\\\text{c}&\text{d}\end{bmatrix}$ is = ad - bc.
View full solution →Assertion: The value of x for which $\begin{vmatrix}3&\text{x}\\\text{x}&1\end{vmatrix}=\begin{vmatrix}3&2\\4&1\end{vmatrix}$ is $\pm2\sqrt{2}.$
Reason: The determinant of a matrix A order 2x2, $\text{A}\begin{bmatrix}\text{a}&\text{b}\\\text{c}&\text{d}\end{bmatrix}$ is = ad - bc.
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.
If $\begin{vmatrix} 3 \text{x}& 7 \\ -2 & 4\\ \end{vmatrix} = \begin{vmatrix} 8 & 7 \\ 6 & 4\\ \end{vmatrix}, $find the value of x.
View full solution →If $\begin{bmatrix} 3 \text{x}& 7 \\ -2 & 4\\ \end{bmatrix} = \begin{bmatrix} 8 & 7 \\ 6 & 4\\ \end{bmatrix}, $find the value of x.
View full solution →If $\begin{bmatrix} \text{x - y }& \text{z} \\ 2\text{x - y }& \text{w} \\ \end{bmatrix} = \begin{bmatrix} -1& 4 \\ 0 & 5\\ \end{bmatrix},$find the value of x + y.
View full solution →$\text{If }x \in \text{N and} \begin{bmatrix} \text{x + 3} & -2 \\ \text{-3x} & \text{2x} \\ \end{bmatrix} = 8, $ then find the value of $x.$
View full solution →What positive value of x makes the following pair of determinants equal? .
$\begin{vmatrix}\text{2x}&3\\5&\text{x} \end{vmatrix}, \begin{vmatrix}\text{16}&3\\5&\text{2} \end{vmatrix}$
View full solution →$\begin{vmatrix}\text{2x}&3\\5&\text{x} \end{vmatrix}, \begin{vmatrix}\text{16}&3\\5&\text{2} \end{vmatrix}$
If $A$ is a non$-$singular symmetric matrix, write whether $A^{-1}$ is symmetric or skew$-$symmetric.
View full solution →Evaluate the determinant:$ \begin{vmatrix}3&-4&5\\1&1&-2\\2&3&1\end{vmatrix}$
View full solution →If $A$ is a square matrix of order $3$ such that $adj\ (2A) = k\ adj\ (A),$ then write the value of $k.$
View full solution →If $A$ is a non$-$singular square matrix such that $|A| = 10,$ find $|A^{-1}|.$
View full solution →Show that points:
A (a, b + c), B (b, c + a), C (c, a + b) are collinear.
Using the properties of determinants, prove that$ \begin{vmatrix} \text{a + b} & \text{b + c} & \text{c + a} \\ \text{b + c} & \text{c + a} & \text{a + b} \\ \text{c + a} & \text{a + b} & \text{b + c} \end{vmatrix}=2 \begin{vmatrix} \text{a} & \text{b} & \text{c} \\ \text{b} & \text{c} & \text{a} \\ \text{c} & \text{a} & \text{b} \end{vmatrix}$.
View full solution →If $\text{A}= \begin{bmatrix} 3 & 1 \\ -1 & 2 \\ \end{bmatrix},$ show the $\text{A}^{2}-\text{5A}+\text{7I}=0$. Hence find $A^{-1}.$
View full solution →Using properties of determinants, prove the following:
$\begin{vmatrix} 3a & -a + b & -a + c \\ a - b & 3b & c - b \\ a - c & b - c & 3c \end{vmatrix} = 3(a + b + c) (ab + bc + ca) $
View full solution →$\begin{vmatrix} 3a & -a + b & -a + c \\ a - b & 3b & c - b \\ a - c & b - c & 3c \end{vmatrix} = 3(a + b + c) (ab + bc + ca) $
If $A = \begin{bmatrix} 2 & -3 & \\ 3 & 4 & \\ \end{bmatrix} $ show that $\text{A^{2} - 6 A + 17 I = 0.}$ Hense find $A^{-1}.$
View full solution →Using properties of determinants, prove the following:$ \begin{vmatrix} a - b -c & 2a & 2a \\ 2b & b- c - a & 2b \\ 2c & 2c & c- a -b \end{vmatrix} = (a + b + c)^{3}$
View full solution →Using properties of determinants, prove that
$\begin{vmatrix} \text{a}^{2} + \text{2a} & \text{2a + 1} & 1 \\ \text{2a + 1} & \text{a + 2} & 1 \\ 3 & 3 & 1 \end{vmatrix} = \text{(a - 1)}^{3}$
View full solution →$\begin{vmatrix} \text{a}^{2} + \text{2a} & \text{2a + 1} & 1 \\ \text{2a + 1} & \text{a + 2} & 1 \\ 3 & 3 & 1 \end{vmatrix} = \text{(a - 1)}^{3}$
Using properties of determinants, prove that:
$\begin{vmatrix} \text{1 + a } & \text{1} & \text{1} \\ \text{1} & \text{1 + b} & \text{1} \\\text{1} & 1 &\text{1 + c} \end{vmatrix}= \text{ abc + bc + ca + ab}$
View full solution →$\begin{vmatrix} \text{1 + a } & \text{1} & \text{1} \\ \text{1} & \text{1 + b} & \text{1} \\\text{1} & 1 &\text{1 + c} \end{vmatrix}= \text{ abc + bc + ca + ab}$
Using properties of determinants, prove that
$\begin{vmatrix} \text{b + c } & \text{c + a} & \text{a + b} \\ \text{q } + \text{r} & \text{r + p} & \text{p + q} \\ \text{y + z} & \text{z + x} &\text{x + y} \end{vmatrix}= \text{2}\begin{vmatrix} \text{a } & \text{b} & \text{c} \\ \text{p} & \text{q} & \text{r} \\ \text{x} & \text{y} &\text{z} \end{vmatrix}$
View full solution →$\begin{vmatrix} \text{b + c } & \text{c + a} & \text{a + b} \\ \text{q } + \text{r} & \text{r + p} & \text{p + q} \\ \text{y + z} & \text{z + x} &\text{x + y} \end{vmatrix}= \text{2}\begin{vmatrix} \text{a } & \text{b} & \text{c} \\ \text{p} & \text{q} & \text{r} \\ \text{x} & \text{y} &\text{z} \end{vmatrix}$
Using properties of determinants, show that $\triangle\text{ABC}$ is isosceles if:
$\begin{vmatrix} 1 & 1 & 1 \\ 1 + \cos\text{A} & 1 + \cos\text{B} & 1 + \cos\text{C} \\ \cos^{2}\text{A} + \cos\text{A} & \cos^{2}\text{B}+\cos\text{B} & \cos^{2}\text{C} + \cos\text{C} \end{vmatrix} = 0 $
View full solution →$\begin{vmatrix} 1 & 1 & 1 \\ 1 + \cos\text{A} & 1 + \cos\text{B} & 1 + \cos\text{C} \\ \cos^{2}\text{A} + \cos\text{A} & \cos^{2}\text{B}+\cos\text{B} & \cos^{2}\text{C} + \cos\text{C} \end{vmatrix} = 0 $
Using properties of determinants, prove that
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Gaurav purchased $5$ pens, $3$ bags and $1$ instrument box and pays $₹ \ 16.$ From the same shop, Dheeraj purchased $2$ pens, 1 bag and $3$ instrument boxes and pays $₹ \ 19,$ while Ankur purchased $1$ pen, $2$ bags and $4$ instrument boxes and pays $₹ \ 25.$
Using the concept of matrices and determinants, answer the following questions.
View full solution →Using the concept of matrices and determinants, answer the following questions.
- The cost of one pen is:
- $₹ \ 2$
- $₹ \ 5$
- $₹ \ 1$
- $₹ \ 3$
- What is the cost of one pen and one bag?
- $₹ \ 3$
- $₹ \ 5$
- $₹ \ 7$
- $₹ \ 8$
- What is the cost of one pen and one instrument box?
- $₹ \ 7$
- $₹ \ 6$
- $₹ \ 8$
- $₹ \ 9$
- Which of the following is correct?
- Determinant is a square matrix.
- Determinant is a number associated to a matrix.
- Determinant is a number associated to a square matrix.
- All of the above.
- From the matrix equation $AB = AC,$ it can be concluded that $B = C$ provided:
- $A$ is singular.
- $A$ is non$-$singular.
- $A$ is symmetric.
- $A$ is square.
The upward speed $v(I)$ of a rocket at time $I$ is approximated by $\text{v}(\text{t})=\text{at}^2+\text{bt}+\text{c},0\leq\text{t}\leq100,$ here $a, b$ and $c$ are constants. It has been found that the speed at $\times t = 3, t = 6$ and $t = 9$ seconds are respectively $64, 133$ and $208$ miles per second..
If $\begin{bmatrix}9&3&1\\36&6&1\\81&9&1\end{bmatrix}^{-1}=\frac{1}{18}\begin{bmatrix}1&-2&1\\-15&24&-9\\54&-54&18\end{bmatrix},$ then answer the following questions.
View full solution →If $\begin{bmatrix}9&3&1\\36&6&1\\81&9&1\end{bmatrix}^{-1}=\frac{1}{18}\begin{bmatrix}1&-2&1\\-15&24&-9\\54&-54&18\end{bmatrix},$ then answer the following questions.
- The value of $b + c$ is:
- $20$
- $21$
- $\frac{3}{4}$
- $\frac{4}{3}$
- The value of $a + c$ is:
- $1$
- $20$
- $\frac{4}{3}$
- None of these.
- $v(t)$ is given by:
- $\text{t}^2+20\text{t}+1$
- $\frac{1}{3}\text{t}^2+20\text{t}+1$
- $\text{t}^2+\frac{1}{3}\text{t}+20$
- $\text{t}^2+\text{t}+1$
- The speed at time $1 = 15 $seconds is:
- $346$ miles/ sec
- $356$ miles/ sec
- $366$ miles/ sec
- $376$ miles/ sec
- The time at which the speed of rocket is $784$ miles/ sec is:
- $20$ seconds
- $30$ seconds
- $25$ seconds
- $27$ seconds
Two schools $A$ and $B$ want to award their selected students on the values of Honesty, Hard work and Punctuality. The school $A$ wants to award $₹\ x$ each, $₹\ y$ each and $₹\ z$ each for the three respective values to its $3, 2$ and $1$ students respectively with a total award money of $₹\ 2200.$ School $B$ wants to spend $₹\ 3100$ to award its $4, 1$ and $3$ students on the respective values $($by giving the same award money to the three values as school $A).$ The total amount of award for one prize on each value is $₹\ 1200.$
Using the concept of matrices and determinants, answer the following questions.
View full solution →Using the concept of matrices and determinants, answer the following questions.
- What is the award money for Honesty?
- $₹\ 350$
- $₹\ 300$
- $₹\ 500$
- $₹\ 400$
- What is the award money for Punctuality?
- $₹\ 300$
- $₹\ 280$
- $₹\ 450$
- $₹\ 500$
- What is the award money for Hard work?
- $₹\ 500$
- $₹\ 400$
- $₹\ 300$
- $₹\ 550$
- If a matrix $P$ is both symmetric and skew$-$symmetric, then $|P|$ is equal to:
- $1$
- $-1$
- $0$
- None of these.
- If P and Qare two matrices such that $PQ = Q$ and $QP = P,$ then $|Q^2|$ is equal to:
- $|Q|$
- $|P|$
- $1$
- $0$
Three shopkeepers Salim, Vijay and Venket are using polythene bags, handmade bags $($prepared by prisoners$)$ and newspaper's envelope as carry bags. It is found that the shopkeepers Salim, Vijay and Venket are using $(20, 30, 40), (30, 40, 20)$ and $(40, 20, 30)$ polythene bags, handmade bags and newspaper's envelopes respectively. The shopkeepers Salim, Vijay and Venket spent $₹\ 250, ₹\ 270$ and $₹\ 200$ on these carry bags respectively.
Using the concept of matrices and determinants, answer the following questions.
View full solution →Using the concept of matrices and determinants, answer the following questions.
- What is the cost of one polythene bag?
- $₹\ 1$
- $₹\ 2$
- $₹\ 3$
- $₹\ 5$
- What is the cost of one handmade bag?
- $₹\ 1$
- $₹\ 2$
- $₹\ 3$
- $₹\ 5$
- What is the cost of one newspaper envelope?
- $₹\ 1$
- $₹\ 2$
- $₹\ 3$
- $₹\ 5$
- Keeping in mind the social conditions, which shopkeeper is better?
- Salim
- Vijay
- Venket
- None of these.
- Keeping in mind the environmental conditions, which shopkeeper is better?
- Salim
- Vijay
- Venket
- None of these.
If there is a statement involving the natural number $n$ such that:
Also, if $A$ is a square matrix of order $n,$ then $A^2$ is defined as $AA.$ In general, $A^m = AA .... A (m$ times$).$
where $m$ is any positive integer. Based on the above information, answer the following questions.
View full solution →- The statement is true for $n = 1$
- When the statement is true for $n = k ($where $k$ is some positive integer$),$ then the statement is also true for $n = k + 1.$
Also, if $A$ is a square matrix of order $n,$ then $A^2$ is defined as $AA.$ In general, $A^m = AA .... A (m$ times$).$
where $m$ is any positive integer. Based on the above information, answer the following questions.
- If $\text{A}=\begin{bmatrix}3&-4\\1&-1\end{bmatrix},$ then for any positive integer $n,$
- $\text{A}^\text{n}=\begin{bmatrix}3\text{n}&-4\text{n}\\\text{n}&-\text{n}\end{bmatrix}$
- $\text{A}^\text{n}=\begin{bmatrix}1+2\text{n}&-4\text{n}\\\text{n}&1-2\text{n}\end{bmatrix}$
- $\text{A}^\text{n}=\begin{bmatrix}3\text{n}&-8\text{n}\\1&-\text{n}\end{bmatrix}$
- $\text{A}^\text{n}=\begin{bmatrix}1+3\text{n}&-4\text{n}\\\text{n}&1-3\text{n}\end{bmatrix}$
- If $\text{A}=\begin{bmatrix}1&2\\0&1\end{bmatrix},$ then $|A^n|,$ where $\text{n}\epsilon\text{ N},$ is equal to:
- $2^n$
- $3^n$
- $n$
- $1$
- If $\text{A}=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ then which of the following holds for all natural numbers $\text{n}\geq1?$
- $A^{n }= nA - (n - 1)I$
- $A^n = 2^{n-1} A - (n - 1)I$
- $A^{n }= nA + (n - 1)I$
- $A^n = 2^{n-1} A + (n - 1)I$
- Let $\text{A}=\begin{bmatrix}\text{a}&0&0\\0&\text{a}&0\\0&0&\text{a}\end{bmatrix}$ and $\text{A}^\text{n}=[\text{a}_{\text{ij}}]_{3\times3}$ for some positive integer $n,$ then the cofactor of $a_{13}$ is:
- $a^n$
- $-a^n$
- $2a^n$
- $0$
- If $A$ is a square matrix such that $|A| = 2,$ then for any positive integer $n, |A^n|$ is equal to:
- $0$
- $2n$
- $2^n$
- $n^2$
Fill in the blanks:
If x, y, z ∈ R, then the value of determinant $\begin{vmatrix}(2^\text{x}+2^{-\text{x}})^2&(2^\text{x}-2^{-\text{x}})^2&1\$3^\text{x}+3^{-\text{x}})^2&(3^\text{x}-3^{-\text{x}})^2&1\$4^\text{x}+4^{-\text{x}})&(4^\text{x}-4^{-\text{x}})^2&1\end{vmatrix}$ is:
View full solution →If x, y, z ∈ R, then the value of determinant $\begin{vmatrix}(2^\text{x}+2^{-\text{x}})^2&(2^\text{x}-2^{-\text{x}})^2&1\$3^\text{x}+3^{-\text{x}})^2&(3^\text{x}-3^{-\text{x}})^2&1\$4^\text{x}+4^{-\text{x}})&(4^\text{x}-4^{-\text{x}})^2&1\end{vmatrix}$ is:
Fill in the blanks:
If $A$ is a matrix of order $3 \times 3,$ then $(A^2)^{-1} =$ ________.
View full solution →If $A$ is a matrix of order $3 \times 3,$ then $(A^2)^{-1} =$ ________.
Fill in the blanks:
$\begin{vmatrix}0&\text{xyz}&\text{x}-\text{z}\\\text{y}-\text{x}&0&\text{y}-\text{z}\\\text{z}-\text{x}&\text{z}-\text{y}&0\end{vmatrix}= ..........$
View full solution →$\begin{vmatrix}0&\text{xyz}&\text{x}-\text{z}\\\text{y}-\text{x}&0&\text{y}-\text{z}\\\text{z}-\text{x}&\text{z}-\text{y}&0\end{vmatrix}= ..........$
Fill in the blanks:
If A is a matrix of order 3 × 3, then |3A| = _______.
If A is a matrix of order 3 × 3, then |3A| = _______.
Fill in the blanks:
If $A$ is invertible matrix of order $3 \times 3,$ then $|A^{-1}|$ _______ .
View full solution →If $A$ is invertible matrix of order $3 \times 3,$ then $|A^{-1}|$ _______ .
State True or False for the statements of the following Exercise:
Let $ \begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=16,$ then $\Delta_1=\begin{vmatrix}\text{p}+\text{x}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}+\text{y}&\text{b} +\text{y}&\text{b}+\text{q}\\\text{r}+\text{z}&\text{c}+ \text{z}&\text{c}+\text{r}\end{vmatrix}=32.$
View full solution →Let $ \begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=16,$ then $\Delta_1=\begin{vmatrix}\text{p}+\text{x}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}+\text{y}&\text{b} +\text{y}&\text{b}+\text{q}\\\text{r}+\text{z}&\text{c}+ \text{z}&\text{c}+\text{r}\end{vmatrix}=32.$
State True or False for the statements of the following Exercise:
If the determinant $\begin{vmatrix}\text{x}+\text{a}&\text{p}+\text{u}&\text{l}+\text{f}\\\text{y}+\text{b}&\text{q}+\text{v}&\text{m}+\text{g}\\\text{z}+\text{c}&\text{r}+\text{w}&\text{n}+\text{h}\end{vmatrix}$ splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.
View full solution →If the determinant $\begin{vmatrix}\text{x}+\text{a}&\text{p}+\text{u}&\text{l}+\text{f}\\\text{y}+\text{b}&\text{q}+\text{v}&\text{m}+\text{g}\\\text{z}+\text{c}&\text{r}+\text{w}&\text{n}+\text{h}\end{vmatrix}$ splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.
State True or False for the statements of the following Exercise:
$\begin{vmatrix}\text{x}+1&\text{x}+2&\text{x}+\text{a}\\\text{x}+2&\text{x}+3&\text{x}+\text{b}\\\text{x}+3&\text{x}+4&\text{x}+\text{c}\end{vmatrix}=0,$ where a, b, c are in A.P.
View full solution →$\begin{vmatrix}\text{x}+1&\text{x}+2&\text{x}+\text{a}\\\text{x}+2&\text{x}+3&\text{x}+\text{b}\\\text{x}+3&\text{x}+4&\text{x}+\text{c}\end{vmatrix}=0,$ where a, b, c are in A.P.
State True or False for the statements of the following Exercise:
The determinant $\begin{vmatrix}\sin\text{A}&\cos\text{A}&\sin\text{A}+\cos\text{B}\\\sin\text{B}&\cos\text{A}&\sin\text{B}+\cos\text{B}\\\sin\text{C}&\cos\text{A}&\sin\text{C}+\cos\text{B}\end{vmatrix}$ is equal to zero.
View full solution →The determinant $\begin{vmatrix}\sin\text{A}&\cos\text{A}&\sin\text{A}+\cos\text{B}\\\sin\text{B}&\cos\text{A}&\sin\text{B}+\cos\text{B}\\\sin\text{C}&\cos\text{A}&\sin\text{C}+\cos\text{B}\end{vmatrix}$ is equal to zero.
State True or False for the statements of the following Exercise:
$\big(\text{aA})^{-1}-\frac{1}{\text{a}}\text{A}^{-1},$ where a is any real number and A is a square matrix.
View full solution →$\big(\text{aA})^{-1}-\frac{1}{\text{a}}\text{A}^{-1},$ where a is any real number and A is a square matrix.
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