TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(a)

Question 1.
Write the following as a single matrix.
(i) [2 1 3] + [0 0 0]
Answer:
[2 1 3] + [0 0 0] = [2 + 0 1 + 0 3 + 0]
= [2 1 3]

(ii) $\left[\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right]+\left[\begin{array}{r} -1 \\ 1 \\ 0 \end{array}\right]$
Answer:
$\left[\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right]+\left[\begin{array}{r} -1 \\ 1 \\ 0 \end{array}\right]$ = $\left[\begin{array}{r} 0-1 \\ 1+1 \\ -1+0 \end{array}\right]$ = $\left[\begin{array}{r} -1 \\ 2 \\ -1 \end{array}\right]$

(iii) $\left[\begin{array}{ccc} 3 & 9 & 0 \\ 1 & 8 & -2 \end{array}\right]+\left[\begin{array}{ccc} 4 & 0 & 2 \\ 7 & 1 & 4 \end{array}\right]$

Answer:

(iv) $\left[\begin{array}{rr} -1 & 2 \\ 1 & -2 \\ 3 & -1 \end{array}\right]+\left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \\ -2 & 1 \end{array}\right]$
Answer:

Question 2.
If A = $\left[\begin{array}{cc} -1 & 3 \\ 4 & 2 \end{array}\right]$, B = $\left[\begin{array}{cc} 2 & 1 \\ 3 & -5 \end{array}\right]$, X = $\left[\begin{array}{ll} \mathbf{x}_1 & \mathbf{x}_2 \\ \mathbf{x}_3 & \mathbf{x}_4 \end{array}\right]$ and A + B = X then find the values of x₁, x₂, x₃ and x₄.

Answer:
A + B = X

⇒ x₁ = 1, x₂ = 4, x₃ = 7, x₄ = – 3.

Question 3.
If A = $\left[\begin{array}{ccc} -1 & -2 & 3 \\ 1 & 2 & 4 \\ 2 & -1 & 3 \end{array}\right]$ B = $\left[\begin{array}{ccc} 1 & -2 & 5 \\ 0 & -2 & 2 \\ 1 & 2 & -3 \end{array}\right]$ and C = $\left[\begin{array}{ccc} -2 & 1 & 2 \\ 1 & 1 & 2 \\ 2 & 0 & 1 \end{array}\right]$ then find A + B + C.

Answer:
A + B + C =

Question 4.
If A = $\left[\begin{array}{ccc} 3 & 2 & -1 \\ 2 & -2 & 0 \\ 1 & 3 & 1 \end{array}\right]$, B = $\left[\begin{array}{ccc} -3 & -1 & 0 \\ 2 & 1 & 3 \\ 4 & -1 & 2 \end{array}\right]$ and X = A + B then find X.

Answer:
X = A + B

Question 5.
If $\left[\begin{array}{cc} x-3 & 2 y-8 \\ z+2 & 6 \end{array}\right]$ = $\left[\begin{array}{cc} 5 & 2 \\ -2 & a-4 \end{array}\right]$ then find the values of x, y, z and a. [May 2006, Mar. 14]

Answer:
Given $\left[\begin{array}{cc} x-3 & 2 y-8 \\ z+2 & 6 \end{array}\right]$ = $\left[\begin{array}{cc} 5 & 2 \\ -2 & a-4 \end{array}\right]$
We have x – 3 = 5, 2y – 8 = 2, z + 2 = – 2, a – 4 = 6
⇒ x = 8, y = 5, z = – 4, a = 10

II.
Question 1.
If $\left[\begin{array}{ccc} x-1 & 2 & 5-y \\ 0 & z-1 & 7 \\ 1 & 0 & a-5 \end{array}\right]$ = $\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 1 & 0 & 0 \end{array}\right]$ then find the values x, y, z and a.
Answer:
Given $\left[\begin{array}{ccc} x-1 & 2 & 5-y \\ 0 & z-1 & 7 \\ 1 & 0 & a-5 \end{array}\right]$ = $\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 1 & 0 & 0 \end{array}\right]$
we have x – 1 = 1, 5 – y = 3, z – 1 = 4,
a – 5 = 0
⇒ x = 2, y = 2, z = 5, a = 5

Question 2.
Find the trace of $\left[\begin{array}{ccc} 1 & 3 & -5 \\ 2 & -1 & 5 \\ 1 & 0 & 1 \end{array}\right]$

Answer:
Trace of the given matrix
= 1 – 1 + 1 = sum of the diagonal elements
= 1

Question 3.
If A = $\left[\begin{array}{ccc} 0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & -6 \end{array}\right]$ and B = $\left[\begin{array}{ccc} -1 & 2 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right]$ find A – B and 4A – 5B.

Answer:

Question 4.
If A = $\left[\begin{array}{lll} 1 & 2 & 3 \\ 3 & 2 & 1 \end{array}\right]$ and B = $\left[\begin{array}{lll} 3 & 2 & 1 \\ 1 & 2 & 3 \end{array}\right]$ find 3B – 2A.

Answer: