Question

Given matrix A `[(4 sin 30°, cos 0°),(cos 0°, 4 sin 30°)]` and B = `[(4),(5)]`. If AX = B.

1. Write the order of matrix X.
2. Find the matrix ‘X’.

Answer 1

A = \[\begin{bmatrix} 4 \sin 30^\circ & \cos0^\circ\\ \cos0^\circ & 4 \sin 30^\circ \end{bmatrix}\]and B = \[{\begin{bmatrix} {4}\\{5}\end{bmatrix}}\]

1. Let the order of matrix X = m × n

Order of matrix A = 2 × 2

Order of matrix B = 2 × 1

Now, AX = B

⇒ Order of matrix X = m × n = 2 × 1

2. Let the matrix X = \[\begin{bmatrix}x\\y\end{bmatrix}\]

AX = B

 ⇒ \[\begin{bmatrix} 4 \sin 30^\circ & \cos0^\circ\\ \cos0^\circ & 4 \sin 30^\circ \end{bmatrix}\] \[\begin{bmatrix}x \\ y\end{bmatrix}\] \[{\begin{bmatrix} {4}\\{5}\end{bmatrix}}\]

⇒\[\begin{bmatrix}4(\frac{1}{2})&1\\1&4(\frac{1}{2})\end{bmatrix}\]\[\begin{bmatrix}x\\y\end{bmatrix}\] = \[\begin{bmatrix}4\\5\end{bmatrix}\]

⇒\[\begin{bmatrix}2&1\\1&2\end{bmatrix}\]\[\begin{bmatrix}x\\y\end{bmatrix}\] = \[\begin{bmatrix}4\\5\end{bmatrix}\]

⇒\[\begin{bmatrix}2x + y\\x + 2y\end{bmatrix}\] = \[\begin{bmatrix}4\\5\end{bmatrix}\]

⇒ 2x + y = 4 ...(i)

And x + 2y = 5 ...(ii)

Subtracting (ii) from (i), we get

⇒ 2x + y – (x + 2y) = 4 – 5

⇒ 2x + y – x – 2y = 4 – 5

x – y = –1 ...(iii)

Adding (i) and (ii), we get

⇒ 2x + y + x + 2y = 4 + 5

⇒ 3x + 3y = 9

⇒ x + y = 3 ...(iv)

Adding (iii) and (iv), we get

2x = 2

⇒ x = 1

Substitute x in (iv), we get y = 2

Hence, the matrix X = \[\begin{bmatrix}1\\2\end{bmatrix}\]