SATHEE: Matrices and Determinants 1 Question 7

Matrices and Determinants 1 Question 7

7. If $A = [\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ c & 2 & b \end{array}]$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where, $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

(2015 Main)

(a) $(2, - 1)$

(b) $(- 2, 1)$

(d) $(- 2, - 1)$

Show Answer

Answer:

Correct Answer: 7. (d)

Solution:

Given, $A = [\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array}], A^{T} = [\begin{array}{ccc} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & - 2 & b \end{array}]$ and

$\begin{aligned} A A^{T} & = [\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array}] [\begin{array}{ccc} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & - 2 & b \end{array}] \\ = [\begin{array}{ccc} 9 & 0 & a + 4 + 2 b \\ 0 & 9 & 2 a + 2 - 2 b \\ a + 4 + 2 b & 2 a + 2 - 2 b & a^{2} + 4 + b^{2} \end{array}] \end{aligned}$

It is given that, $A A^{T} = 9 I$

$\begin{matrix} \Rightarrow [\begin{array}{ccc} 9 & 0 & a + 4 + 2 b \\ 0 & 9 & 2 a + 2 - 2 b \\ a + 4 + 2 b & 2 a + 2 - 2 b & a^{2} + 4 + b^{2} \end{array}] = 9 [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] \\ \Rightarrow [\begin{array}{ccc} 9 & 0 & a + 4 + 2 b \\ 0 & 9 & 2 a + 2 - 2 b \\ a + 4 + 2 b & 2 a + 2 - 2 b & a^{2} + 4 + b^{2} \end{array}] = [\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}] \end{matrix}$

On comparing, we get

$a + 4 + 2 b = 0 \Rightarrow a + 2 b = - 4$ …(i)

$2 a + 2 - 2 b = 0 \Rightarrow a - b = - 1$ …(ii)

and $a^{2} + 4 + b^{2} = 9$ …(iii)

On solving Eqs. (i) and (ii), we get

$a = - 2, b = - 1$

This satisfies Eq. (iii)

Hence, $(a, b) = (- 2, - 1)$

Matrices and Determinants 1 Question 7

7. If $A = [\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ c & 2 & b \end{array}]$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where, $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 1 Question 7

7. If A=[12221−2c2b] is a matrix satisfying the equation AAT=9I, where, I is 3×3 identity matrix, then the ordered pair (a,b) is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

7. If $A = [\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ c & 2 & b \end{array}]$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where, $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to