1For \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), which of these is \( A^{-1} \): \( \frac{1}{|A|} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), \( \frac{1}{|A|} \begin{bmatrix} a & -b \\ -c & d \end{bmatrix} \), \( \frac{1}{|A|} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \) or \( \frac{1}{|A|} \begin{bmatrix} -d & b \\ c & -a \end{bmatrix} \)?Show Answer

2In which case is the inverse \( A^{-1} \) of a matrix A not defined: when A is a singular matrix, when A is a non-singular matrix, when A is a square matrix, or when A is an identity matrix?Show Answer

3Which of these is a property of the inverse matrix: \( (A^{T})^{T} = A \), \( (A^{-1})^{-1} = A \), \( (AB)^{T} = B^{T}A^{T} \) or \( AI = IA = A \)?Show Answer

4If \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), which of these is \( A^{-1} \): \( \frac{1}{5} \begin{bmatrix} -2 & 3 \\ 1 & -4 \end{bmatrix} \), \( \frac{1}{5} \begin{bmatrix} 2 & -3 \\ -1 & 4 \end{bmatrix} \), \( \frac{1}{5} \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix} \) or \( \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} \)?Show Answer

5Multiply \( A = \begin{bmatrix} 5 & 1 \\ 2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix} \) and check whether they are inverses of each other.Show Answer

6Multiply \( A = \begin{bmatrix} 5 & -3 \\ -3 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix} \) and prove that they are inverses of each other.Show Answer

7Multiply \( A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix} \) and prove that they are inverses of each other.Show Answer

8Multiply \( A = \begin{bmatrix} 4 & 7 \\ 5 & 9 \end{bmatrix} \) and \( B = \begin{bmatrix} 9 & -7 \\ -5 & 4 \end{bmatrix} \) and prove that they are inverses of each other.Show Answer

9Find the inverse of \( A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \).Show Answer

10Find the inverse of \( B = \begin{bmatrix} 7 & 4 \\ 3 & 2 \end{bmatrix} \).Show Answer

11Find the inverse of \( P = \begin{bmatrix} 2 & 0 \\ 3 & 1 \end{bmatrix} \).Show Answer

12Find the inverse of \( R = \begin{bmatrix} \cos A & -\sin A \\ \sin A & \cos A \end{bmatrix} \).Show Answer

13If \( \begin{bmatrix} a & 1 \\ 5 & 2 \end{bmatrix} \) and \( \begin{bmatrix} 2 & -1 \\ -5 & b \end{bmatrix} \) are inverses of each other, find a and b.Show Answer

14If \( \begin{bmatrix} p & 2p - 9 \\ -q & 3 \end{bmatrix} \) and \( \begin{bmatrix} 3 & 5 \\ q & p \end{bmatrix} \) are inverses of each other, find p and q.Show Answer

15The inverse of a matrix A is \( A^{-1} = \begin{bmatrix} 3 & 2 \\ 5 & 4 \end{bmatrix} \). Find the matrix A.Show Answer

16If \( A = \begin{bmatrix} 2 & 0 \\ 5 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \), find \( |A| \) and \( |B| \).Show Answer

17For the same \( A = \begin{bmatrix} 2 & 0 \\ 5 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \), find \( A^{-1} \) and \( B^{-1} \).Show Answer

18For the same A and B, find \( (AB)^{-1} \).Show Answer

19For the same A and B, check that \( (AB)^{-1} = B^{-1}A^{-1} \).Show Answer

20Why must \( ad - bc \) not be zero for \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) to have an inverse? What happens when \( ad - bc = 0 \)?Show Answer

21Does \( \begin{bmatrix} 6 & 3 \\ 4 & 2 \end{bmatrix} \) have an inverse? Give a reason.Show Answer

22For \( A = \begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix} \) and \( B = \begin{bmatrix} -2 & 7 \\ -3 & 9 \end{bmatrix} \), is \( (AB)^{-1} \) equal to \( A^{-1}B^{-1} \)?Show Answer

23Find the inverse of \( \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} \) and check your answer.Show Answer