List of top Mathematics Questions on Matrices asked in PSEB XII

If a matrix \( A \) is symmetric as well as skew-symmetric, then \( A = 0 \).

Solve by Matrix Method
Given: \[ x + 2y - 3z = 6, \quad 3x + 2y - 2z = 3, \quad 2x - y + z = 2 \]

Express the matrix
\[ \begin{bmatrix} 7 & 0 & 3 \\ 2 & 4 & 1 \\ -5 & 6 & 8 \end{bmatrix} \] as the sum of symmetric and skew-symmetric matrices.

If \[ A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}, \] prove that \[ A^2 - 4A - 5I = 0. \] Hence, find \( A^{-1} \).

Find \( (AB)^{-1} \), if

\[ A = \begin{bmatrix} 3 & 4 \\ 1 & 1 \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \]

Verify \( (AB)^T = B^T A^T \).

Given:

\[ A = \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix} \]

If \( A \) is an invertible matrix of order \( n \), then \( \text{adj} (\text{adj} A) = |A|^{n-2} A \).

A 2x2 matrix whose elements are \( a_{ij} = \frac{(t + n)^2}{2} \) is:

The adjoint of the matrix

\[ \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \]

is:

If

\[ \begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix} \]

then \( x \) is equal to:

If

\[ \begin{bmatrix} 1 & 4 \\ -3 & x \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 8 & -3 \end{bmatrix} \]

then the value of \( x \) is:

The matrix

\[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

is:

If

\[ A = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix}, \quad f(x) = x^2 + 2x - 3 \]

then find \( f(A) \).

Express

\[ \begin{bmatrix} 5 \\ 3 \end{bmatrix} \]

as the sum of a symmetric matrix and a skew-symmetric matrix.

If

\[ X = \begin{bmatrix} 3 & 4 \\ 2 & -1 \end{bmatrix}, \quad 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

then find the matrix \( Y \).

If

\[ A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 5 & 0 \end{bmatrix} \]

then verify that \( (AB)^T = B^T A^T \).

If \( AB = C \) where \( B \) and \( C \) are of order \( 3 \times 5 \), then the order of \( A \) is:

If \( A \) is an invertible matrix, then \( \det(A^{-1}) \) is:

If \( A \) is a non-singular matrix of order 3 and \( |A| = 2 \), then \( |\text{adj}(A)| \) equals:

Solve the following system of linear equations by matrix method: \[ 5x + y - 3z = -10, \quad 3x - 2y + z = -3, \quad x + 3y + z = 4 \]

If the order of the matrix \( A \) is \( 3 \times 2 \), then the order of matrix \( A^T \) is:

If \( A \) is a square matrix of order \( 2 \times 2 \) and \( |A| = 5 \), then \( | \text{Adj}(A) | \) is:

If \( A = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} \) and \( f(x) = x^2 - 10x + 13 \), then show that \( f(A) = O \) and using this result find \( A^{-1} \).

Express \( \begin{bmatrix} 5 & 1 \\ 3 & 7 \end{bmatrix} \) as the sum of a symmetric matrix and a skew-symmetric matrix.