1For \( A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -3 \\ -2 & -5 \end{bmatrix} \), what are the orders of A and B?
2How can the matrix formed by multiplying A by B be found?
3For the same A and B, are AB and BA equal?
4Matrix A has m rows and n columns and matrix B has n rows and p columns, so AB can be found. The number of columns of B is p and the number of rows of A is m. Is it possible to find BA?
5In which situation is the multiplication of two matrices possible?
6If three matrices A, B and C satisfy \( (AB)C = A(BC) \), what is this property called?
7If three matrices A, B and C satisfy \( A(B + C) = AB + AC \), what is this property called?
8If A is a square matrix and I is the identity matrix of the same order, and \( AI = IA = A \), what is this property called?
9If \( P = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) and \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), what is the product PI?
10If \( W = \begin{bmatrix} -3 \\ 4 \\ 5 \end{bmatrix} \) and \( R = [3 \;\; 2 \;\; 1] \), what is the order of the matrix WR?
11Find the product, if possible: \( [5 \;\; 4] \begin{bmatrix} 1 \\ 3 \end{bmatrix} \)
12Find the product, if possible: \( \begin{bmatrix} 1 \\ 3 \end{bmatrix} [5 \;\; 4] \)
13Find the product, if possible: \( [3 \;\; 2 \;\; 1] \begin{bmatrix} -3 \\ 4 \\ 5 \end{bmatrix} \)
14Find the product, if possible: \( \begin{bmatrix} 0 & 1 \\ 5 & -4 \\ 4 & 5 \end{bmatrix} \begin{bmatrix} 3 & 4 & 7 \\ 2 & -1 & 10 \end{bmatrix} \)
15Find the product, if possible: \( \begin{bmatrix} -1 & 5 \\ 3 & 7 \end{bmatrix} \begin{bmatrix} 6 & 1 \\ 8 & 2 \end{bmatrix} \)
16Find the product, if possible: \( \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} -3 & 4 \\ 0 & 6 \\ 1 & 2 \end{bmatrix} \)
17If \( Q = \begin{bmatrix} 3 & 6 \\ -2 & -4 \end{bmatrix} \), find \( Q^2 \).
18If \( M = \begin{bmatrix} -2 & 3 \\ -1 & 2 \end{bmatrix} \) and \( N = \begin{bmatrix} 4 & -3 \\ 5 & 0 \end{bmatrix} \), find MN and NM.
19If \( C = \begin{bmatrix} 3 & 5 \\ 0 & 1 \end{bmatrix} \), \( D = \begin{bmatrix} -2 & 4 \\ -1 & 3 \end{bmatrix} \) and \( E = \begin{bmatrix} 2 & -3 \\ 6 & 4 \end{bmatrix} \), find C(D + E).
20If \( A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \), prove that \( A^2 = 2A \).
21If \( X = \begin{bmatrix} 1 & -1 \\ 3 & -3 \end{bmatrix} \) and \( Y = \begin{bmatrix} 2 & -5 \\ 2 & -5 \end{bmatrix} \), prove that XY is a zero matrix.
22If \( R = \begin{bmatrix} 2 & -3 \\ 1 & -2 \end{bmatrix} \), prove that \( R^2 = I \), where I is the identity matrix of order \( 2 \times 2 \).
23If \( A = \begin{bmatrix} 2 & 1 \\ 4 & -1 \end{bmatrix} \), prove that \( A^2 - A - 6I = O \), where I and O are the identity matrix and the zero matrix of order \( 2 \times 2 \).
24If \( U = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix} \), \( V = \begin{bmatrix} p & q \\ 0 & r \end{bmatrix} \) and \( UV = U + V \), find the values of p, q and r.
25If \( X = \begin{bmatrix} 6 & 3 \\ -2 & 0 \end{bmatrix} \) and \( Y = \begin{bmatrix} 2 & 5 \\ 1 & -4 \end{bmatrix} \), prove that \( (XY)^T = Y^T.X^T \).
26If \( A = \begin{bmatrix} 0 & 2 \\ 1 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix} \), prove that \( AB \neq BA \).
27With \( A = \begin{bmatrix} 0 & 2 \\ 1 & 3 \end{bmatrix} \), \( B = \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix} \) and \( C = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix} \), prove that \( A(BC) = (AB)C \).
28With the same A, B and C, prove that \( A(B + C) = AB + AC \).
29With \( A = \begin{bmatrix} 0 & 2 \\ 1 & 3 \end{bmatrix} \) and \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), prove that \( IA = AI = A \).
30With \( A = \begin{bmatrix} 0 & 2 \\ 1 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix} \), prove that \( (AB)^T = B^T.A^T \).
31A is a \( 2 \times 3 \) matrix and B is a \( 3 \times 4 \) matrix. What is the order of AB? Can BA be found?
32Why can only a square matrix be squared?
33With \( A = \begin{bmatrix} 1 & 2 \\ 0 & 9 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix} \), show that \( (AB)^T \) is not equal to \( A^T.B^T \).