Sample QuestionsDeterminants and Matrices questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $A=\left[\begin{array}{cc}-2 & 1 \\ 0 & 3\end{array}\right]$ and $f(x)=2 x^2-3 x$, then $f(A)=$
- A
$\left[\begin{array}{cc}14 & 1 \\ 0 & -9\end{array}\right]$
- B
$\left[\begin{array}{cc}-14 & 1 \\ 0 & 9\end{array}\right]$
- ✓
$\left[\begin{array}{cc}14 & -1 \\ 0 & 9\end{array}\right]$
- D
$\left[\begin{array}{cc}-14 & -1 \\ 0 & -9\end{array}\right]$
Answer: C.
View full solution →For suitable matrices A, B, the false statement is ___________
- ✓
$(A B)^{\top}=A^{\top} B^{\top}$
- B
$\left(A^{\top}\right)^{\top}=A$
- C
$(A-B)^{\top}=A^{\top}-B^{\top}$
- D
$(A+B)^{\top}=A^{\top}+B^{\top}$
Answer: A.
View full solution →If $\left[\begin{array}{cc}x & 3 x-y \\ z x+z & 3 y-w\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right]$, then
- ✓
x = 3, y = 7, z = 1, w = 14
- B
x = 3, y = -5, z = -1, w = -4
- C
x = 3, y = 6, z = 2, w = 7
- D
x = -3, y = -7, z = -1, w = -14
Answer: A.
View full solution →If $A+B=\left[\begin{array}{ll}7 & 4 \\ 8 & 9\end{array}\right]$ and $A-B=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$, then the value of $A$ is
- A
$\left[\begin{array}{ll}3 & 1 \\ 4 & 3\end{array}\right]$
- ✓
$\left[\begin{array}{ll}4 & 3 \\ 4 & 6\end{array}\right]$
- C
$\left[\begin{array}{ll}6 & 2 \\ 8 & 6\end{array}\right]$
- D
$\left[\begin{array}{cc}7 & 6 \\ 8 & 12\end{array}\right]$
Answer: B.
View full solution →If $\left[\begin{array}{ll}5 & 7 \\ x & 1 \\ 2 & 6\end{array}\right]-\left[\begin{array}{cc}1 & 2 \\ -3 & 5 \\ 2 & y\end{array}\right]=\left[\begin{array}{cc}4 & 5 \\ 4 & -4 \\ 0 & 4\end{array}\right]$, then
Answer: C.
View full solution →If $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 2 & 6\end{array}\right], B=\left[\begin{array}{ccc}1 & 0 & -2 \\ 3 & -1 & 4\end{array}\right]$, find $A B^{\top}$ and $A^{\top} B$.
View full solution →Find $x, y, z$ if $\left\{\left[\begin{array}{lll}1 & 3 & 2 \\ 2 & 0 & 1 \\ 3 & 1 & 2\end{array}\right]+2\left[\begin{array}{lll}3 & 0 & 2 \\ 1 & 4 & 5 \\ 2 & 1 & 0\end{array}\right]\right\}\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$
View full solution →Find $x, y, z$ if $\left\{\left[\begin{array}{ll}0 & 1 \\ 1 & 0 \\ 1 & 1\end{array}\right]-3\left[\begin{array}{cc}2 & 1 \\ 3 & -2 \\ 1 & 3\end{array}\right]\right\}\left[\begin{array}{l}2 \\ 1\end{array}\right]=\left[\begin{array}{c}x-1 \\ y+1 \\ 2 z\end{array}\right]$
View full solution →If $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, show that $(A+B)(A-B) \neq A^2-B^2$.
View full solution →If $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]$, show that $A-4 A+3 I=0$.
View full solution →Prove The Theorem : If $\mathrm{A}\left(x_1, y_1\right), \mathrm{B}\left(x_2, y_2\right)$ and $\mathrm{C}\left(x_3, y_3\right)$ are vertices of triangle $A B C$ then the area of triangle is
$
\frac{1}{2}\left|\begin{array}{lll}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array}\right|
$
View full solution →If $A=\left[\begin{array}{cc}2 & -4 \\ 3 & -2 \\ 0 & 1\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 2 \\ -2 & 1 & 0\end{array}\right]$, show that $(A B)^{\top}=B^{\top} A^{\top}$.
View full solution →If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0\end{array}\right]$, show that $A B$ and $B A$ are both singularmatrices.
View full solution →Show that the lines x – y = 6, 4x – 3y = 20 and 6x + 5y + 8 = 0 are concurrent. Also, find the point of concurrence.
Find the area of quadrilateral whose vertices are A(0, -4), B(4, 0), C(-4,0), D (0, 4).
If A = diag[2, -3, -5], B = diag[4, -6, -3] and C = diag [-3, 4, 1], then find : (i) B + C – A ,(ii)2A + B – 5C.
Solve the following linear equations by Cramer’s Rule. x + y + 2z = 7, 3x + 4y – 5z = 5, 2x – y + 3z = 12
Solve the following linear equations by Cramer’s Rule. $2 x+3 y+3 z=5, x-2 y+z=-4,3 x-y-2 z=3$
View full solution →Solve the following linear equations by Cramer’s Rule. $\frac{1}{x}+\frac{1}{y}=\frac{3}{2}, \quad \frac{1}{y}+\frac{1}{z}=\frac{5}{6}, \quad \frac{1}{z}+\frac{1}{x}=\frac{4}{3}$
View full solution →Solve the following linear equations by Cramer’s Rule. 2x – y + z = 1, x + 2y + 3z = 8, 3x + y – 4z = 1
Prove The Theorem : The necessary condition for the equation $a_1 x+b_1 y+c_1=0, a_z x+b_y y+c_2=0$, $a_y x+b_y y+c_3=0$ to be consistent is
$
\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|=0
$
View full solution →Prove that $\left|\begin{array}{lll}1 & a & b c \\ 1 & b & c a \\ 1 & c & a b\end{array}\right|=\left|\begin{array}{lll}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{array}\right|$
View full solution →Two farmers Shantaram and Kantaram cultivate three crops rice, wheat, and groundnut. The sale (in Rupees) of these crops by both the farmers for the month of April and may 2008 is given below,
Find
(i) the total sale in rupees for two months of each farmer for each crop.
(ii) the increase in sales from April to May for every crop of each farmer.
If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$, prove that $\mathrm{A}^n=\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right]$, for all $n \in \mathbb{N}$.
View full solution →If $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$, prove that $A^n=\left[\begin{array}{cc}1+2 n & -4 n \\ n & 1-2 n\end{array}\right]$, for all $n \in \mathbf{N}$.
View full solution →Find $x, y$ if $\left\{-1\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]+3\left[\begin{array}{lll}2 & -3 & 7 \\ 1 & -1 & 3\end{array}\right]\right\}\left[\begin{array}{c}5 \\ 0 \\ -1\end{array}\right]=\left[\begin{array}{l}x \\ y\end{array}\right]$
View full solution →Find x, y if,$\begin{aligned} & {\left[\begin{array}{lll}0 & -1 & 4\end{array}\right]\left\{2\left[\begin{array}{cc}4 & 5 \\ 3 & 6 \\ 2 & -1\end{array}\right]+3\left[\begin{array}{cc}4 & 3 \\ 1 & 4 \\ 0 & -1\end{array}\right]\right\}} \\ & =\left[\begin{array}{ll}x & y\end{array}\right] .\end{aligned}$
View full solution →If $A=\left[\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right]$, find $A^3$.
View full solution →If $A=\left[\begin{array}{cc}-3 & 2 \\ 2 & -4\end{array}\right], B=\left[\begin{array}{cc}1 & x \\ y & 0\end{array}\right]$ and $(A+B)(A-B)=A^2-B^2$, find $x$ and $y$.
View full solution →If $A=\left[\begin{array}{ccc}4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3\end{array}\right]$, show that $A^2=1$.
View full solution →