Question:

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

Updated On: Sep 9, 2025
  • (2,-1)
  • (-2,1)
  • (2,1)
  • (-2,-1)
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The Correct Option is D

Solution and Explanation

$AAT = 9\, I$
$\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{bmatrix}\begin{bmatrix}1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b\end{bmatrix}=\begin{bmatrix}9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9\end{bmatrix}$
$a+4+2 \,b=0 $
$\Rightarrow a+2 b=-4\,\,\,\,\,\,\,\,...,(i)$
$2 a+2-2\, b=0$
$ \Rightarrow a-b=-1\,\,\,\,\,\,\,\,\,...(ii)$
From i and ii
$3\, b=-3 $
$\Rightarrow b=-1$
$a=-2$
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Concepts Used:

Matrices

Matrix:

A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.

The basic operations that can be performed on matrices are:

  1. Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
  2. Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
  3. Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication. 
  4. Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal. 
  5. Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.