Question:
How many $3 \times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is $5$ ?
How many $3 \times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is $5$ ?
Updated On: Jun 14, 2022
- 126
- 198
- 162
- 135
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The Correct Option is B
Solution and Explanation
Let matrix $M =\left[ k _{ i } j \right]$
Then sum of diagonal entries $=\sum k _{ ij }^{2}$
$\Rightarrow \sum k _{ ij }^{2}=5$
where $k _{ ij }$ are from $\{0,1,2\}$
$\Rightarrow$ Total number of matrices $={ }^{9} C _{5}+{ }^{9} C _{1} \cdot{ }^{8} C _{1}=198$
Then sum of diagonal entries $=\sum k _{ ij }^{2}$
$\Rightarrow \sum k _{ ij }^{2}=5$
where $k _{ ij }$ are from $\{0,1,2\}$
$\Rightarrow$ Total number of matrices $={ }^{9} C _{5}+{ }^{9} C _{1} \cdot{ }^{8} C _{1}=198$
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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.