NCERT Solutions
Class 12 - Mathematics Part-1 - Chapter 3: MATRICES

Miscellaneous Exercise on Chapter 3

If \(A\) and \(B\) are symmetric matrices, prove that \(AB - BA\) is a skew symmetric matrix.

Show that the matrix \(B'AB\) is symmetric or skew symmetric according as \(A\) is symmetric or skew symmetric.

Find the values of \(x, y, z\) if the matrix

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

satisfy the equation \(A'A = I\).

For what values of \(x\) :

\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix}\begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = 0\)?

If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), show that \(A^2 - 5A + 7I = 0\).

Find \(x\), if

\(\begin{bmatrix} x & -5 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}\begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = 0\).

A manufacturer produces three products \(x, y, z\) which he sells in two markets. Annual sales are indicated below:

Market I: \(x = 10000,\; y = 2000,\; z = 18000\)

Market II: \(x = 6000,\; y = 20000,\; z = 8000\)

(a) If unit sale prices of \(x, y\) and \(z\) are \(\text{₹}2.50,\; \text{₹}1.50\) and \(\text{₹}1.00\), respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are \(\text{₹}2.00,\; \text{₹}1.00\) and 50 paise respectively. Find the gross profit.

Find the matrix \(X\) so that

\(X\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}\).

Choose the correct answer.

If \(A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is such that \(A^2 = I\), then

(A) \(1 + \alpha^2 + \beta\gamma = 0\)

(B) \(1 - \alpha^2 + \beta\gamma = 0\)

(C) \(1 - \alpha^2 - \beta\gamma = 0\)

(D) \(1 + \alpha^2 - \beta\gamma = 0\)

Choose the correct answer.

If the matrix \(A\) is both symmetric and skew symmetric, then

(A) \(A\) is a diagonal matrix

(B) \(A\) is a zero matrix

(C) \(A\) is a square matrix

(D) None of these

Choose the correct answer.

If \(A\) is a square matrix such that \(A^2 = A\), then \((I + A)^3 - 7A\) is equal to

(A) \(A\)

(B) \(I - A\)

(C) \(I\)

(D) \(3A\)