Question

Suppose \(A B=0,\) where \(A\) and \(B\) are square matrices. Show that: a. If one of \(A\) and \(B\) has an inverse, the other is zero. b. It is impossible for both \(A\) and \(B\) to have inverses. c. \((B A)^{2}=0\)

Short answer

If one matrix has an inverse, the other is zero; both cannot be invertible; \\( (BA)^2 = 0 \\\.

Step-by-step solution

Understanding the Product of Matrices Equaling Zero

We are given that the product of two square matrices \(AB = 0\). This means that when matrix \(A\) is multiplied by matrix \(B\), the result is the zero matrix. Our task is to explore the implications of this equation when considering the invertibility of the matrices.

Prove Part (a) - Invertibility Implies the Other Matrix is Zero

Assume \(A\) is invertible. This means there exists a matrix \(A^{-1}\) such that \(A^{-1}A = I\), where \(I\) is the identity matrix. Multiply both sides of \(AB = 0\) by \(A^{-1}\) to get \(A^{-1}AB = A^{-1} 0\). This simplifies to \(B = 0\) since \(A^{-1}A = I\). Similarly, if \(B\) is invertible, the same argument shows that \(A = 0\). Thus, if one matrix is invertible, the other must be the zero matrix.

Prove Part (b) - Both Matrices Cannot Be Invertible

Assume for contradiction that both \(A\) and \(B\) are invertible. If \(A\) is invertible, the same argument as in Step 2 leads to \(B = 0\). But \(B\) cannot be the zero matrix if it is invertible, as zero matrices are not invertible. Therefore, it's impossible for both \(A\) and \(B\) to be invertible simultaneously.

Prove Part (c) - \\( (BA)^2 = 0 \\\

From \(AB = 0\), it follows that \(B(AB) = B0 = 0\). Since \(AB = 0\), \((BA)(BA) = BA(BA) = B(AB)A = B0A = 0\). Thus, \((BA)^2 = 0\) because multiplying a matrix with a zero matrix results in a zero matrix.

Key concepts

Invertible Matrices

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that possesses an inverse. In simple terms, if a matrix is invertible, there exists another matrix such that when the two are multiplied, the result is an identity matrix.
This identity matrix is a special type of square matrix where all the diagonal elements are 1, and all other elements are 0. It functions much like the number 1 does in arithmetic - acting as a neutral element in multiplication.
Invertibility is an important property because it suggests that the matrix can be "undone" or reversed in some sense. If you have a matrix \(A\) and its inverse \(A^{-1}\), then \(A \times A^{-1} = I\), where \(I\) is the identity matrix, and this operation leaves the identity matrix unchanged.

• A matrix is invertible if its determinant is not zero.
• If a matrix is not invertible, it is called singular.
• For a matrix \(A\) to have an inverse, there must be a matrix \(A^{-1}\) such that \(A \times A^{-1} = A^{-1} \times A = I\).

Zero Matrix

The zero matrix is a matrix in which all the elements are zero. It acts like the number 0 does in basic arithmetic when it comes to matrix operations.
When a matrix is multiplied by a zero matrix, or a zero matrix by a matrix, the result is a zero matrix.
This behavior is similar to how multiplying any number by zero results in zero. A zero matrix can have any dimensions, provided all elements are zero.

• For any matrix \(A\), \(A \times 0 = 0\) and \(0 \times A = 0\).
• The zero matrix is a crucial element in matrix equations like \(AB = 0\), where it represents the outcome of such multiplication.
• The zero matrix itself is never invertible, as there is no matrix that can be multiplied by a zero matrix to result in an identity matrix.

Matrix Multiplication

Matrix multiplication involves computing the product of two matrices, and it is a fundamental operation in linear algebra.
Unlike regular multiplication, matrix multiplication is not commutative, meaning \(A \times B eq B \times A\) in general.
The product of matrices \(A\) and \(B\) is calculated by taking rows from \(A\) and columns from \(B\), multiplying the corresponding elements, and summing the results.

• The number of columns in the first matrix must match the number of rows in the second matrix in order for the multiplication to be possible.
• The resulting matrix has dimensions determined by the number of rows in the first matrix and the number of columns in the second matrix.
• In cases where the product \(AB = 0\), the zero matrix result signifies that one or both of the matrices may include inherent zero properties or dependency.

Understanding these properties aids in evaluating conditions like \(AB = 0\), where even if both matrices themselves might not be zero, their interaction yields a zero matrix.