### Question 1. Find the order of the following matrices.

### 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,

[highlight]$ c = 4 $[/highlight]

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

[highlight]$ a = -4 $[/highlight]

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

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

$ \Rightarrow 2b = 4 – 7 $

$ \Rightarrow 2b = -3 $

[highlight]$ \Rightarrow b = {-3 \over 2} $[/highlight]

From equation (4), we have,

$ 4d – 2d = 6 $

$ \Rightarrow 2d = 6 $

[highlight]$ \Rightarrow d = 3 $[/highlight]

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