# Worksheet on Matrix Multiplication

Practice the questions given in the Worksheet on Matrix Multiplication.

1. Let A = $$\begin{bmatrix} -10 & 1\\ 3 & -2 \end{bmatrix}$$, B = $$\begin{bmatrix} 6\\ -7 \end{bmatrix}$$. Find AB and BA if possible.

2. Let A = $$\begin{bmatrix} 1 & -1\\ 3 & 4 \end{bmatrix}$$, B = $$\begin{bmatrix} 0 & 1\\ 2 & -3 \end{bmatrix}$$.

(i) Find AB and BA if possible.

(ii) Verify if AB = BA.

(iii) Find A2.

(iv) Find AB2.

3. If A = $$\begin{bmatrix} sin \, \, 30^{\circ} + cos \, \, 60^{\circ} & tan \, \, 45^{\circ} - cot \, \, 45^{\circ}\\ cos \, \, 90^{\circ} & sin \, \, 90^{\circ} \end{bmatrix}$$ then prove that A3 = A2 =A.

4. If A = $$\begin{bmatrix} cos \, \, \theta & -sin \, \, \theta\\ sin \, \, \theta & cos \, \, \theta \end{bmatrix}$$ and B = $$\begin{bmatrix} cos \, \, \theta & sin \, \, \theta\\ -sin \, \, \theta & cos \, \, \theta \end{bmatrix}$$, then prove that AB = I, where I is the unit matrix.

5. Let A = $$\begin{bmatrix} -2 & 9\\ 1 & 3 \end{bmatrix}$$, B = $$\begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}$$ and C = $$\begin{bmatrix} -1 & 2\\ 3 & -1 \end{bmatrix}$$.

(i) Find (AB)C.

(ii) Prove that A(BC) = (AB)C.

1. AB = $$\begin{bmatrix} -67\\ 32 \end{bmatrix}$$; BA is not possible because number of columns in B ≠ number of rows in A

2. (i) AB = $$\begin{bmatrix} -2 & 4\\ 8 & -9 \end{bmatrix}$$; B = $$\begin{bmatrix} 3 & 4\\ -7 & -14 \end{bmatrix}$$

(ii) AB ≠ BA.

(iii) $$\begin{bmatrix} -2 & -5\\ 15 & 13 \end{bmatrix}$$

(iv) $$\begin{bmatrix} 8 & -14\\ -18 & 35 \end{bmatrix}$$

5. (i) $$\begin{bmatrix} 14 & 7\\ 8 & 4 \end{bmatrix}$$