Class 12 Matrices Important Questions

1. If \( A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \), prove by the Principle of Mathematical Induction that \( A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \) for all \( n \in \mathbb{N} \).

2. Find the value of \( \alpha \) for which the matrix \( A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix} \) satisfy the relation \( A^2 = B \).

3. If \( A \) and \( B \) are square matrices of the same order, prove that \( (A + B)^2 = A^2 + AB + BA + B^2 \). Under what specific condition does the algebraic identity \( (A + B)^2 = A^2 + 2AB + B^2 \) hold true for matrices?

4. Let \( A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \), \( B = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix} \), and \( C = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix} \). Find a matrix \( D \) such that \( CD – AB = O \).

5. If \( A \) is a square matrix such that \( A^2 = A \), then show that \( (I + A)^3 – 7A = I \).

6. Show that the matrix \( A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \) satisfies the equation \( A^2 – 4A – 5I = O \). Hence, find \( A^{-1} \) using this result.

7. If \( A \) and \( B \) are symmetric matrices of the same order, prove that \( AB \) is symmetric if and only if \( A \) and \( B \) commute, i.e., \( AB = BA \).

8. Find the value of \( x \) for which the following product is a null matrix:

\[ \begin{bmatrix} x & 4 & 1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & x \end{bmatrix} \begin{bmatrix} x \\ 4 \\ -1 \end{bmatrix} = O \]

9. If \( A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \), find \( k \) such that \( A^2 = 8A + kI \), where \( I \) is the identity matrix of order 2.

10. Prove that every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.

11. Using elementary column transformations, find the inverse of the matrix \( A = \begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix} \).

12. If \( A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix} \) are such that \( (A + B)^2 = A^2 + B^2 \), find the values of \( a \) and \( b \).

13. If \( A \) is an orthogonal matrix (i.e., \( AA^T = I \)), prove that \( A^T \) is also orthogonal and that \( A^{-1} \) exists and is equal to \( A^T \).

14. Show that the matrix \( B^T AB \) is symmetric or skew-symmetric according as \( A \) is symmetric or skew-symmetric.

15. Using elementary row transformations, find the inverse of the following \( 3 \times 3 \) matrix:

\[ A = \begin{bmatrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{bmatrix} \]