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
- (2,-1)
- (-2,1)
- (2,1)
- (-2,-1)
The Correct Option is D
Solution and Explanation
$\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{bmatrix}\begin{bmatrix}1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b\end{bmatrix}=\begin{bmatrix}9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9\end{bmatrix}$
$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:
- 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.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- 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.
- 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.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.