Let A and B be two \(3 × 3\) non-zero real matrices such that AB is a zero matrix. Then
- the system of linear equations AX = 0 has a unique solution
- the system of linear equations AX = 0 has infinitely many solutions
- B is an invertible matrix
- adj(A) is an invertible matrix
The Correct Option is B
Solution and Explanation
AB is zero matrix
\(⇒ |A| = |B| = 0\)
Hence, neither A nor B is invertible
If \(|A| = 0\)
\(⇒ |adj A| = 0\) so adj A is not invertible
\(AX = 0\) is homogeneous system and \(|A| = 0\)
Therefore, it is having infinitely many solutions.
So, the correct option is (B): the system of linear equations \(AX = 0\) has infinitely many solutions
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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.