Question

For three square matrix \(A, B \& C\) if \(A B C=O\) \& \(|A| \neq 0, B\) is non-zero singular matrix then (1) \(\mathrm{C}\) must be zero matrix (2) C must be non-singular matrix (3) C must be singular matrix (4) Data insufficient

Short answer

C must be singular matrix (3).

Step-by-step solution

Understand given conditions

Three matrices A, B, and C are given such that the product of all three matrices ABC equals the zero matrix O. Additionally, the determinant of A is not zero (meaning A is non-singular or invertible), and B is a non-zero singular matrix (meaning it is non-invertible and has a determinant of zero).

Analyze the non-singularity of A

Since \(|A| eq 0\), matrix A is invertible. When multiplying matrices, if A were multiplied by another matrix, the resultant matrix properties will largely depend on the properties of the other matrices in the product.

Analyze the singularity of B

Matrix B is singular, meaning it has a determinant of zero. This implies that B has no inverse and has at least one zero eigenvalue. Hence, B will map at least one non-zero vector to the zero vector.

Evaluate the product ABC

The fact that ABC = O means the resulting product of these matrices is the zero matrix. Given that A is invertible, multiplying (A^-1) on both sides of ABC = O yields BCL = O where L is some arbitrary matrix. Since B is singular, this condition means that regardless of matrix C, the product can result in the zero matrix only if C must be singular.

Conclusion

Given the conditions, it is determined that matrix C must be singular for the equation ABC = O to hold true. Any matrix multiplied by a singular matrix that results in a zero matrix implies that the other matrices must also be aligned in such structure, pointing towards a singular nature for matrix C.

Key concepts

Singular Matrix

A singular matrix is one that does not have an inverse. This occurs if and only if its determinant is zero. In simpler terms, if you try to find a matrix that, when multiplied by a given singular matrix, results in the identity matrix, you will fail.
Here are some key points about singular matrices:

• They are non-invertible.
• Their rows or columns are linearly dependent, meaning one row or column can be expressed as a combination of others.
• They have at least one zero eigenvalue.

Given these points, if you encounter a problem involving singular matrices, you already know that certain operations, like finding inverses, are off the table.
This was critical in the given exercise because matrix B was described as singular.
This singularity was key to understanding that matrix C must also be structured to make ABC equal to the zero matrix (O).

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces a new matrix. The process is not straightforward like regular arithmetic multiplication but involves a systematic approach:

• The element in the resulting matrix at position (i, j) is obtained by multiplying the elements of the i-th row of the first matrix with the elements of the j-th column of the second matrix and summing the products.
• The number of columns in the first matrix must equal the number of rows in the second matrix.

In our exercise, we dealt with the product of three matrices, ABC, that resulted in the zero matrix, O.
Because matrix A was invertible, multiplying both sides of the equation by its inverse helped us analyze the roles of B and C further.
Ultimately, since B was singular, its interaction within the multiplication ensured that certain properties of matrix C were vital for maintaining the equation.

Determinant

The determinant of a matrix is a special number that can tell you a lot about the matrix. It’s calculated from the elements of the matrix in a specific manner. Here are some essential points on determinants:

• If the determinant is zero, the matrix is singular (non-invertible).
• If the determinant is non-zero, the matrix is non-singular (invertible).
• The determinant helps in finding whether a system of linear equations has a unique solution.

In the given exercise, the determinant of matrix A was not zero, which meant A was non-singular and invertible.
This allowed us to manipulate the matrix equation effectively, helping us deduce that matrix C had to be singular for the entire product ABC to result in the zero matrix, O.
Understanding the determinant was pivotal in solving for the properties of matrix C.