Let S be the set containing all 3 × 3 matrices with entries from {-1, 0, 1}. The total number of matrices A ∈ S such that the sum of all the diagonal elements of AT A is 6 is ________.
Correct Answer: 5376
Solution and Explanation
The correct answer is 5376
Sum of all diagonal elements is equal to sum of square of each element of the matrix.
\(i.e., A =\) \(\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix}\)
then \(t_r(A.A^T)\)
\(= a^{2}_{1}+a^{2}_{2}+a^{2}_{3}+b^{2}_{1}+b^{2}_{2}+b^{2}_{3}+c^{2}_{1}+c^{2}_{2}+c^{2}_{3}\)
\(∵ a_i, b_i, c_i ∈{-1,0,1} \)for \(i = 1,2,3\)
∴ Exactly three of them are zero and rest are 1 or – 1.
Total number of possible matrices
\(^9C_3×2^6\)
\(= \frac{9×8×7}{6}×64\)
= 5376
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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.