If A = 2 - 4 - 5 3 then find A + A ' and A A ' .

Given, A = 2 - 4 - 5 3 ⇒ A ' = 2 − 4 − 5 3 T = 2 − 5 − 4 3 And, A + A ' = 2 − 4 − 5 3 + 2 − 5 − 4 3 = 2 + 2 − 4 − 5 − 5 − 4 3 + 3 = 4 − 9 − 9 6 Now, A A ' = 2 - 4 - 5 3 2 - 5 - 4 3 = 4 + 16 - 10 - 12 - 10 - 12 25 + 9 = 20 - 22 - 22 34

Is 0 1 4 − 1 0 7 − 4 − 7 0 Symmetric or skew symmetric?

Let A = 0 1 4 − 1 0 7 − 4 − 7 0 Consider, A ' = 0 1 4 − 1 0 7 − 4 − 7 0 ' = 0 − 1 − 4 1 0 − 7 4 7 0 = − 0 1 4 − 1 0 7 − 4 − 7 0 = - A Since, A ' = - A So, A is Skew symmetric matrix.

If A = cos ⁡ α sin ⁡ α − sin ⁡ α cos ⁡ α , show that A A ' = A ' A = I .

We have, A = cos ⁡ α sin ⁡ α − sin ⁡ α cos ⁡ α Now, A ' = cos ⁡ α sin ⁡ α − sin ⁡ α cos ⁡ α T = cos ⁡ α − sin ⁡ α sin ⁡ α cos ⁡ α Consider, A ⋅ A ' = cos ⁡ α sin ⁡ α − sin ⁡ α cos ⁡ α cos ⁡ α − sin ⁡ α sin ⁡ α cos ⁡ α = cos 2 ⁡ α + sin 2 ⁡ α − sin ⁡ α cos ⁡ α + sin ⁡ α cos ⁡ α − sin ⁡ α cos ⁡ α + cos ⁡ α sin ⁡ α sin 2 ⁡ α + cos 2 ⁡ α = 1 0 0 1 ∵ sin 2 θ + cos 2 θ = 1 = I . . . 1 Consider, A ' · A = cos ⁡ α − sin ⁡ α sin ⁡ α cos ⁡ α cos ⁡ α sin ⁡ α − sin ⁡ α cos ⁡ α = cos 2 ⁡ α + sin 2 ⁡ α cos ⁡ α sin ⁡ α − sin ⁡ α cos ⁡ α sin ⁡ α cos ⁡ α − cos ⁡ α sin ⁡ α sin 2 ⁡ α + cos 2 ⁡ α = 1 0 0 1 ∵ sin 2 θ + cos 2 θ = 1 = I . . . 2 From 1 and 2 we get the desired result.

If A = 1 5 3 2 4 0 3 − 1 − 5 and B = 2 − 1 0 0 − 2 5 1 2 0 then find 3 A − 4 B ' .

Given: A = 1 5 3 2 4 0 3 − 1 − 5 and B = 2 − 1 0 0 − 2 5 1 2 0 So, B ' = 2 − 1 0 0 − 2 5 1 2 0 ' = 2 0 1 − 1 − 2 2 0 5 0 Consider, 3 A - 4 B ' = 3 1 5 3 2 4 0 3 − 1 − 5 − 4 2 0 1 − 1 − 2 2 0 5 0 = 3 15 9 6 12 0 9 − 3 − 15 − 8 0 4 − 4 − 8 8 0 20 0 = 3 − 8 15 − 0 9 − 4 6 + 4 12 + 8 0 − 8 9 − 0 − 3 − 20 − 15 − 0 = − 5 15 5 10 20 − 8 9 − 23 − 15

If A = 7 − 2 − 1 2 5 3 and B = − 2 − 1 4 2 − 1 0 then find A B ' and B A ' .

We have, B = − 2 − 1 4 2 − 1 0 ⇒ B ' = − 2 − 1 4 2 − 1 0 ' = − 2 4 − 1 − 1 2 0 Consider, A B ' = 7 − 2 − 1 2 5 3 − 2 4 − 1 − 1 2 0 = − 14 + 2 28 − 4 − 7 + 0 2 − 2 − 4 + 4 1 + 0 − 10 − 3 20 + 6 − 5 + 0 = − 12 24 − 7 0 0 1 − 13 26 − 5 Now, A = 7 − 2 − 1 2 5 3 So, A ' = 7 − 2 − 1 2 5 3 ' = 7 − 1 5 − 2 2 3 Consider, B A ' = − 2 − 1 4 2 − 1 0 7 − 1 5 − 2 2 3 = − 14 + 2 2 − 2 − 10 − 3 28 − 4 − 4 + 4 20 + 6 − 7 + 0 1 + 0 − 5 + 0 = − 12 0 − 13 24 0 26 − 7 1 − 5

E M B I B E

Intermediate First Year Mathematics Paper 1A>Algebra, Vector Algebra, Trigonometry>Chapter 3 - Matrices>Exercise 3 (c)

Andhra Pradesh Board Solutions for Exercise 3: Exercise 3 (c)

Author:Andhra Pradesh Board

Andhra Pradesh Board Mathematics Solutions for Exercise - Andhra Pradesh Board Solutions for Exercise 3: Exercise 3 (c)

Attempt the free practice questions from Exercise 3: Exercise 3 (c) with hints and solutions to strengthen your understanding. Intermediate First Year Mathematics Paper 1A solutions are prepared by Experienced Embibe Experts.

Questions from Andhra Pradesh Board Solutions for Exercise 3: Exercise 3 (c) with Hints & Solutions

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q I

If $A = [\begin{matrix} 2 & - 4 \\ - 5 & 3 \end{matrix}]$ then find $A + A' and A A'$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q I

If $A = [\begin{array}{ccc} - 1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & x & 7 \end{array}]$ is a symmetric matrix, then find $x$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q I

If $A = [\begin{array}{ccc} 0 & 2 & 1 \\ - 2 & 0 & - 2 \\ - 1 & x & 0 \end{array}]$ is a skew symmetric matrix, then find $x$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q I

Is $[\begin{array}{ccc} 0 & 1 & 4 \\ - 1 & 0 & 7 \\ - 4 & - 7 & 0 \end{array}]$ Symmetric or skew symmetric?

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q II

If $A = [\begin{array}{ll} \cos α \sin α \\ - \sin α \cos α \end{array}]$ , show that $A A' = A' A = I$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q II

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

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1A>Chapter 3 - Matrices>Exercise 3 (c)>Q II

If $A = [\begin{array}{cc} 7 & - 2 \\ - 1 & 2 \\ 5 & 3 \end{array}]$ and $B = [\begin{array}{cc} - 2 & - 1 \\ 4 & 2 \\ - 1 & 0 \end{array}]$ then find $A B' and B A'$ .