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

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

Class 12


Miscellaneous Exercise

1. If A and B are symmetric matrices, prove that AB–BA is a skew symmetric matrix.

2. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

3. 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 \(A’A=I\).

4. For what values of x: \(A=\begin{bmatrix}1&2&1\end{bmatrix}\) \(\begin{bmatrix}1&2&0 \\2&0&1\\1&0&2 \end{bmatrix}\)\(\begin{bmatrix}0\\2\\x \end{bmatrix}=O?\)

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

6. Find x, if \(A=\begin{bmatrix}x&-5&-1\end{bmatrix}\) \(\begin{bmatrix}1&0&2 \\0&2&1\\2&0&3 \end{bmatrix}\)\(\begin{bmatrix}x\\4\\1 \end{bmatrix}=O\).

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

(a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 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 Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

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

9. If \(A=\begin{bmatrix}α&β \\ γ &-α\end{bmatrix}\) is such that \(A^2=I\), then

(A) \(1+α^2+βγ=0\)
(B) \(1-α^2+βγ=0\)
(C) \(1-α^2-βγ=0\)
(D) \(1+α^2-βγ=0\).

10. 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

11. If A is square matrix such that \(A^2=A\), then \((I+A)^3–7A\) is equal to
(A) A
(B) I–A
(C) I
(D) 3A