Matrix Properties When AB is Null

This question concerns the properties of two square matrices, A and B, which have the same order. We are given a specific condition: their product, AB, equals the null matrix (\(O\), a matrix where all elements are zero). We need to determine the correct consequence among the given options.

Key Matrix Types Definitions

• Null Matrix: A matrix where all entries are zero. For example, the \(2 \times 2\) null matrix is represented as \(O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\).
• Singular Matrix: A square matrix is classified as singular if its determinant is equal to zero (det(M) = 0). A key characteristic of singular matrices is that they do not have a multiplicative inverse.
• Non-singular Matrix: Conversely, a square matrix is non-singular if its determinant is not zero (det(M) \(\neq\) 0). Non-singular matrices possess a unique multiplicative inverse, denoted as \(M^{-1}\).

Condition AB = O Analysis

We are given that A and B are square matrices of the same order, and their product \(AB = O\).

Let's focus on the scenario where matrix A is non-singular. By definition, a non-singular matrix has a multiplicative inverse, \(A^{-1}\). This property allows us to manipulate the equation \(AB = O\).

We can apply the multiplicative inverse property as follows:

1. Start with the given equation: \(AB = O\)
2. Multiply both sides of the equation by the inverse of A, \(A^{-1}\), specifically from the left side: \(A^{-1}(AB) = A^{-1}O\)
3. Using the associative property of matrix multiplication, we can regroup the terms on the left side: \((A^{-1}A)B = A^{-1}O\)
4. We know that the product of a matrix and its inverse yields the identity matrix (\(I\)), so \(A^{-1}A = I\). Also, the product of any matrix (including the null matrix) and the null matrix is the null matrix: \(A^{-1}O = O\). Thus, the equation becomes: \(IB = O\)
5. Multiplying any matrix by the identity matrix (\(I\)) leaves the matrix unchanged (\(IB = B\)). Therefore, the equation simplifies to: \(B = O\)

This derivation shows that if matrix A is non-singular, matrix B must be the null matrix (\(O\)) for the condition \(AB = O\) to hold true.

Evaluating Matrix Product Options

Option 1 Analysis: Both Null Matrices

This statement is not necessarily correct. Consider a non-singular matrix A (e.g., \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\), where det(A) = 4-6 = -2 \(\neq\) 0) and the null matrix B (\(B = O\)). The product \(AB = AO = O\). In this case, only B is the null matrix, not necessarily both A and B.

Option 2 Analysis: Either Null Matrix

This statement is not always true. It's possible for the product \(AB\) to be the null matrix even when neither A nor B is the null matrix. For example, let \(A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\). Both matrices are singular (det(A) = 0, det(B) = 0) and neither is the null matrix. However, their product is \(AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O\).

Option 3 Analysis: Non-Singular A Implies Null B

This statement is correct, as proven in the "Condition AB = O Analysis" section above. If A is non-singular, its inverse \(A^{-1}\) exists. Multiplying the equation \(AB = O\) by \(A^{-1}\) on the left directly leads to the conclusion that \(B\) must be the null matrix (\(O\)).

Option 4 Analysis: Both Singular Matrices

This statement is not necessarily true. As demonstrated in the analysis for Option 3, A can be non-singular while B is the null matrix (\(O\), which is also a singular matrix). Therefore, it's not required for *both* matrices to be singular.

Conclusion: Matrix Product Implications

When the product of two square matrices A and B of the same order results in the null matrix (\(AB = O\)), it implies certain conditions. The most definitive conclusion is that if one of the matrices, say A, is non-singular, then the other matrix, B, must inevitably be the null matrix (\(O\)).