New Arrivals

# Exercise 1.1 (Math 9)

### Solution:

$$A = \begin {bmatrix} 2 & 3 \\ -5 & 6 \\ \end{bmatrix}$$
Order of matrix A is 2-by-2

$$B = \begin {bmatrix} 2 & 0 \\ 3 & 5 \\ \end{bmatrix}$$
Order of matrix B is 2-by-2

$$C = \begin {bmatrix} 2 & 4 \\ \end{bmatrix}$$
Order of matrix C is 1-by-2

$$D = \begin {bmatrix} 4 \\ 0 \\ 6 \\ \end{bmatrix}$$
Order of matrix D is 3-by-1

$$E = \begin {bmatrix} a & d \\ b & e \\ c & f \\ \end{bmatrix}$$
Order of matrix E is 3-by-2

$$F = \begin {bmatrix} 2 \\ \end{bmatrix}$$
Order of matrix F is 1-by-1

$$G = \begin {bmatrix} 2 & 3 & 0 \\ 1 & 2 & 3 \\ 2 & 4 & 5 \\ \end{bmatrix}$$
Order of matrix G is 3-by-3

$$H = \begin {bmatrix} 2 & 3 & 4 \\ 1 & 0 & 6 \\ \end{bmatrix}$$
Order of matrix H is 2-by-3

### Question 2. Which of the following matrices are equal?

$A = \begin {bmatrix} 3 \\ \end{bmatrix},$       $B = \begin {bmatrix} 3 & 5 \\ \end{bmatrix},$        $C = \begin {bmatrix} 5 – 2 \\ \end{bmatrix},$       $D = \begin {bmatrix} 5 & 3 \\ \end{bmatrix},$       $E = \begin {bmatrix} 4 & 0 \\ 6 & 2 \\ \end{bmatrix},$       $F = \begin {bmatrix} 2 \\ 6 \\ \end{bmatrix},$       $G = \begin {bmatrix} 3 – 1 \\ 3 + 3 \\ \end{bmatrix},$       $H = \begin {bmatrix} 4 & 0 \\ 6 & 2 \\ \end{bmatrix},$       $I= \begin {bmatrix} 3 & 3 + 2 \\ \end{bmatrix},$       $J = \begin {bmatrix} 2 + 2 & 2 – 2 \\ 2 + 4 & 2 + 0 \\ \end{bmatrix}$

### Solution:

The matrices A and C are equal as the order of  matrix A = order of matrix C and the corresponding elements of both matrices are equal.

The matrices B and I are equal as the order of  matrix B = order of matrix I and the corresponding elements of both matrices are equal.

The matrices E, H and J are equal as the order of  matrix E = order of matrix H = order of matrix J and the corresponding elements of these three matrices are equal.

The matrices F and G are equal as the order of  matrix F = order of matrix G and the corresponding elements of both matrices are equal.

### Question 3. Find the values of a, b, c and d which satisfy the matrix equation.

$\begin {bmatrix} a + c & a + 2b \\ c – 1 & 4d – 6 \\ \end{bmatrix} = \begin {bmatrix} 0 & -7 \\ 3 & 2d \\ \end{bmatrix}$

### Solution:

As the matrices at both sides of given equation are equal so their corresponding elements must be equal.  Therefore,

$a + c = 0 \;\;\;\;\; (1)$

$a + 2b = -7 \;\;\;\;\; (2)$

$c – 1 = 3 \;\;\;\;\; (3)$

$4d – 6 = 2d \;\;\;\;\; (4)$

From equation (3), we have,

$c = 4$

By putting $c = 4$ in equation (1), we have

$a = -4$

Now putting $a = -4$ in equation (2), we have

$-4 + 2b = -7 \;\;\;\;\; (2)$

$\Rightarrow 2b = 4 – 7$

$\Rightarrow 2b = -3$

$\Rightarrow b = {-3 \over 2}$

From equation (4), we have,

$4d – 2d = 6$

$\Rightarrow 2d = 6$

$\Rightarrow d = 3$