Sample QuestionsMatrices (p-1) questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $A =\left[\begin{array}{ll}\alpha & 4 \\ 4 & \alpha\end{array}\right]$ and $\left| A ^3\right|=729$ then $\alpha=$
Answer: C.
View full solution →If a $3 \times 3$ matrix $B$ has its inverse equal to $B$, then $B^2=$
- A
$\left[\begin{array}{lll}0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]$
- B
$\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1\end{array}\right]$
- C
$\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right]$
- ✓
$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
Answer: D.
View full solution →If $A^2+m A+n l=O$ and $n \neq 0,|A| \neq 0$, then $A^{-1}=$ ________
- A
$\frac{-1}{m}(A+n l)$
- ✓
$\frac{-1}{n}(A+m l)$
- C
$\frac{-1}{m}(I+m A)$
- D
$( A + mnl )$
Answer: B.
View full solution →If $A =$ diag. $\left[ d _1, d _2, d _3, \ldots, d _{ n }\right]$, where $d _1 \neq 0$, for $i =1,2,3, \ldots \ldots ., n$, then $A ^{-1}=$
- ✓
$\operatorname{diag}\left[1 / d _1, 1 / d _2, 1 / d _3, \ldots, 1 / d _{ n }\right]$
- B
- C
- D
$O$
Answer: A.
View full solution →Adjoint of $\left[\begin{array}{ll}2 & -3 \\ 4 & -6\end{array}\right]$ is
- ✓
$\left[\begin{array}{ll}-6 & 3 \\ -4 & 2\end{array}\right]$
- B
$\left[\begin{array}{cc}6 & 3 \\ -4 & 2\end{array}\right]$
- C
$\left[\begin{array}{cc}-6 & -3 \\ 4 & 2\end{array}\right]$
- D
$\left[\begin{array}{cc}-6 & 3 \\ 4 & -2\end{array}\right]$
Answer: A.
View full solution →$A=\left[\begin{array}{cc}2 & 1 \\ 10 & 5\end{array}\right]$ is invertible matrix.
View full solution →Singleton matrix is only row matrix.
If $A$ and $B$ are conformable for the product $AB$, then $(AB)^T = A^TB^T$.
View full solution →If A and B are square matrices of same order, then $(A + B)^2 = A^2 + 2AB + B^2$.
View full solution →If AB and BA both exist, then AB = BA.
If $A =\left[\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right]$ and $A ^{-1}=\left[\begin{array}{cc}1 & -1 \\ x & 2\end{array}\right]$, then $x =$__________
View full solution →$\left( A ^{\top}\right)^{\top}=$____________
View full solution →If $A=\left[a_{i j}\right]_{m \times m}$ is non-singular matrix, then $A^{-1}=\frac{1}{\ldots \ldots} \operatorname{adj}(A)$.
View full solution →If $A=\left[\begin{array}{cc}3 & -5 \\ 2 & 5\end{array}\right]$, then cofactor of $a _{12}$ is____________
View full solution →If $A =\left[ a _{ ij }\right]_{2 \times 3}$ and $B =\left[ b _{ ij }\right]_{ m \times 1}$, and $AB$ is defined, then $m =$____________
View full solution →Find $x, y, z$ if $\left[\begin{array}{lll}2 & x & 5 \\ 3 & 1 & z \\ y & 5 & 8\end{array}\right]$ is a symmetric matrix.
View full solution →Find $k _{\text {, if }}\left[\begin{array}{ll}7 & 3 \\ 5 & k\end{array}\right]$ is a singular matrix.
View full solution →Apply the given elementary transformation on each of the following matrices : $\left[\begin{array}{ccc}3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3\end{array}\right] 3 R _2$ and $C _2 \rightarrow C _2-4 C _1$
View full solution →Apply the given elementary transformation on each of the following matrices : $\left[\begin{array}{cc}2 & 4 \\ 1 & -5\end{array}\right], C _1 \leftrightarrow C _2$
View full solution →Apply the given elementary transformation on each of the following matrices : $\left[\begin{array}{cc}3 & -4 \\ 2 & 2\end{array}\right], R_1 \leftrightarrow R_2$
View full solution →If $A=\left[\begin{array}{ll}2 & 5 \\ 3 & 7\end{array}\right], B=\left[\begin{array}{cc}1 & 7 \\ -3 & 0\end{array}\right]$, find the matrix $A-4 B+7 I$, where I is the unit matrix of order 2 .
View full solution →Check whether following matrices are invertible or not: $\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 5 \\ 2 & 4 & 6\end{array}\right]$
View full solution →Check whether following matrices are invertible or not: $\left[\begin{array}{lll}3 & 4 & 3 \\ 1 & 1 & 0 \\ 1 & 4 & 5\end{array}\right]$
View full solution →Check whether following matrices are invertible or not: $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
View full solution →Check whether following matrices are invertible or not: $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
View full solution →If $A=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$, then show that $A^2-4 A+3l=0$.
View full solution →If $A=\left[\begin{array}{ll}3 & 1 \\ 1 & 5\end{array}\right], B=\left[\begin{array}{cc}1 & 2 \\ 5 & -2\end{array}\right]$, verify $|A B|=|A||B|$.
View full solution →Solve the following equations by method of reduction $: 2x + y = 5, 3x – 5y = -3$
View full solution →Find inverse of the following matrices (if they exist) by elementary transformation : $\left[\begin{array}{ll}2 & 1 \\ 7 & 4\end{array}\right]$
View full solution →Find inverse of the following matrices (if they exist) by elementary transformation : $\left[\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right]$
View full solution →If $A=\left[\begin{array}{cc}-3 & 2 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{ll}1 & a \\ b & 0\end{array}\right]$ and $(A+B)(A-B)=A^2-B^2$, find $a$ and $b$.
View full solution →If $A=\left[\begin{array}{cc}2 & -3 \\ 3 & -2 \\ -1 & 4\end{array}\right], B=\left[\begin{array}{ccc}-3 & 4 & 1 \\ 2 & -1 & -3\end{array}\right]$ verify
(i) $\left( A +2 B ^{\top}\right)^{\top}= A ^{\top}+2 B$
(ii) $\left(3 A -5 B ^{\top}\right)^{\top}=3 A ^{\top}-5 B$
View full solution →If $A =\left[\begin{array}{ll}1 & 5 \\ 7 & 8 \\ 9 & 5\end{array}\right], B =\left[\begin{array}{cc}2 & 4 \\ 1 & 5 \\ -8 & 6\end{array}\right], C =\left[\begin{array}{cc}-2 & 3 \\ 1 & -5 \\ 7 & 8\end{array}\right]$ then show that $( A + B )+ C = A +( B + C )$.
View full solution →Solve the following equations by method of reduction : $x + 2y + z = 3, 3x – y + 2z = 1$ and $2x – 3y + 3z = 2$
View full solution →Find $x, y, z$ if $\left\{5\left[\begin{array}{ll}0 & 1 \\ 1 & 0 \\ 1 & 1\end{array}\right]-\left[\begin{array}{cc}2 & 1 \\ 3 & -2 \\ 1 & 3\end{array}\right]\right\}\left[\begin{array}{l}2 \\ 1\end{array}\right]=\left[\begin{array}{c}x-1 \\ y+1 \\ 2 z\end{array}\right]$
View full solution →If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0\end{array}\right]$ thenshow that $A B$ and $BA$ are both singular matrices.
View full solution →The sum of three numbers is $6$. If we multiply the third number by $3$ and add it to the second number, we get $11$. By adding first and third numbers we get a number that is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.
View full solution →Solve the following equations by method of reduction : $x – 3y + z = 2, 3x + y + z = 1$ and $5x + y + 3z = 3.$
View full solution →Solve the following equations by method of inversion : $x-y+z=4,2 x+y-3 z=0$ and $x+y+z=2$
View full solution →Solve the following equations by method of inversion : $x+y-z=2, x-2 y+z=3$ and $2 x-y-3 z=-1$
View full solution →