Matrix and Operations of Matrices

1 Mark Questions

1. If <math>2 \begin{vmatrix} 3 & 4 \\ 5 & x \end{vmatrix} + \begin{vmatrix} 1 & y \\ 0 & 1 \end{vmatrix} = \begin{vmatrix} 7 & 0 \\ 10 & 5 \end{vmatrix}</math>, then find

1. y).

Delhi 2014

Given, <math>2 \begin{vmatrix} 3 & 4 \\ 5 & x \end{vmatrix} + \begin{vmatrix} 1 & y \\ 0 & 1 \end{vmatrix} = \begin{vmatrix} 7 & 0 \\ 10 & 5 \end{vmatrix}</math> <math display="block">\Rightarrow \begin{vmatrix} 6 & 8 \\ 10 & 2x \end{vmatrix} + \begin{vmatrix} 1 & y \\ 0 & 1 \end{vmatrix} = \begin{vmatrix} 7 & 0 \\ 10 & 5 \end{vmatrix}</math> <math display="block">\Rightarrow \begin{vmatrix} 7 & 8+y \\ 10 & 2x+1 \end{vmatrix} = \begin{vmatrix} 7 & 0 \\ 10 & 5 \end{vmatrix}</math>

On comparing the corresponding elements, we get

<math>8 + y = 0</math> and <math>2x + 1 = 5</math> ⇒ <math>y = -8 \text{ and } x = \frac{5-1}{2} = 2</math> <math>x - y = 2 - (-8) = 10</math> (1) 2. Solve the following matrix equation for x.

<math>\begin{bmatrix} x & 1 \end{bmatrix} \begin{vmatrix} 1 & 0 \\ -2 & 0 \end{vmatrix} = 0</math> Delhi 2014

We have, <math>\begin{bmatrix} x & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = 0</math>

By using matrix multiplication, we get

<math>[x-2 \ 0] = [0 \ 0]</math>

On comparing the corresponding elements from both sides, we get

<math>x-2=0 \Rightarrow x=2</math> (1) 3. If A is a square matrix such that <math>A^2 = A</math>, then write the value of <math>7A - (I + A)^3</math>, where I is an identity matrix. All India 2014

We have, <math>A^2 = A</math>

Now,

<math>7A - (I + A)^3 = 7A - [I^3 + A^3 + 3IA(I + A)]</math> <math display="block">[::(x+y)^3 = x^3 + y^3 + 3xy(x+y)]</math> <math>= 7A - [I + A^2 \cdot A + 3A(I + A)]</math> [:: <math>I^3 = I</math>] <math>= 7A - [I + A \cdot A + 3AI + 3A^{2}] [:: A^{2} = A, given]</math> <math>= 7A - [I + A + 3A + 3A]</math> [:: <math>AI = A</math>] <math>= 7A - [I + 7A] = -I</math> (1)

4. If <math>\begin{vmatrix} x-y & z \\ 2x-y & w \end{vmatrix} = \begin{vmatrix} -1 & 4 \\ 0 & 5 \end{vmatrix}</math>, then find the value of <math>x + y</math>. All India 2014

We have, <math>\begin{vmatrix} x-y & z \\ 2x-y & w \end{vmatrix} = \begin{vmatrix} -1 & 4 \\ 0 & 5 \end{vmatrix}</math>

On comparing the corresponding elements, we get

<math>x - y = -1</math>

...(i) and <math>2x - y = 0</math>

...(ii)

On solving the above equations, we get and <math>x = 1</math> <math>y = 2</math>

Now, <math>x + y = 1 + 2 = 3</math> (1)