Matrices and Determinants - Class 11 (HSSC-I) Mathematics

Q1

$\begin{bmatrix} 2 & 3 & 1 \\ 1 & -1 & 2 \\ 3 & 2 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ can be transformed into a system of linear equations as

Q2

The determinant of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & 9 \end{bmatrix}$ is equal to the product of elements of

Q3

Which of the following values of $\lambda$ will make the matrix $\begin{bmatrix} 3 & 8 & \lambda \\ 1 & 2-\lambda & 1 \\ 0 & 1 & 0 \end{bmatrix}$ singular?

Q4

Using the matrix $A = \begin{bmatrix} 2 & 0 & -4 \\ 4 & -2 & -2 \\ -4 & 2 & 6 \end{bmatrix}$, the cofactor element of $A_{32}$ is

Q5

A matrix $A$ is defined as $A = \begin{bmatrix} 2 & 0 & -4 \\ 4 & -2 & -2 \\ -4 & 2 & 6 \end{bmatrix}$. The minor element of $a_{23}$ is

Q6

If $4Z$ is a scalar matrix then, $Z$ is a/an

Q7

$\begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}$ is equal to

Q8

A square matrix is defined as $M=[m_{ij}]$, where $i$ is the number of rows and $j$ is the number of columns.
If matrix $M$ is a diagonal matrix, then
I. $m_{ij} = 0$ for $i \neq j$
II. $m_{ij} = 0$ for $i = j$
III. $m_{ij} \neq 0$ for $i = j$

Q9

It is given that $\begin{vmatrix} 5 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{vmatrix} = 5$, $\begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \end{vmatrix} = 2$, $\begin{vmatrix} 5 & 1 & 1 \\ 0 & 0 & 2 \\ 0 & -1 & 1 \end{vmatrix} = 10$, and $\begin{vmatrix} 5 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{vmatrix} = -5$. On solving the equation $\begin{bmatrix} 5 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$, the value of $y$ will be

Q10

The determinant $\begin{vmatrix} 3 & 6 & 3m \\ 1 & 2 & m \\ ? & ? & ? \end{vmatrix}$

Q11

If $\begin{vmatrix} a & b & c \\ b & a & b \\ c & c & a \end{vmatrix} = x$, then $\begin{vmatrix} a & -3b & c \\ b & -3a & b \\ c & -3c & a \end{vmatrix}$ will be equal to

Q12

If $A = \begin{bmatrix} p & 1 & -p \\ 2 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$ is a singular matrix, then the value of $p$ will be

Q13

The determinant of the matrix $A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & -1 & 1 \\ 3 & -2 & -2 \end{bmatrix}$ is

Q14

For the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & x \\ 2 & 3 & 1 \end{bmatrix}$, the minor of the element at the position of $x$