Question

By considering the matrices $$ \mathrm{A}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad \mathrm{B}=\left(\begin{array}{ll} 0 & 0 \\ 3 & 4 \end{array}\right) $$ show that \(A B=0\) does not imply that either \(A\) or \(B\) is the zero matrix but that it does imply that at least one of them is singular.

Short answer

Neither \(A\) nor \(B\) is the zero matrix, but both are singular.

Step-by-step solution

- Understand the Given Matrices

Identify the matrices provided:\[A = \begin{pmatrix}1 & 0 \0 & 0\end{pmatrix}, \quad B = \begin{pmatrix}0 & 0 \3 & 4\end{pmatrix}\]

- Perform the Matrix Multiplication

Calculate the product of matrices \(A\) and \(B\):\[AB = \begin{pmatrix}1 & 0 \ 0 & 0\end{pmatrix} \begin{pmatrix}0 & 0 \ 3 & 4\end{pmatrix} = \begin{pmatrix}(1\cdot 0 + 0\cdot 3) & (1\cdot 0 + 0\cdot 4) \ (0\cdot 0 + 0\cdot 3) & (0\cdot 0 + 0\cdot 4)\end{pmatrix} = \begin{pmatrix}0 & 0 \ 0 & 0\end{pmatrix}\]

- Examine Matrices for Zero Elements

Notice that neither \(A\) nor \(B\) is the zero matrix since \(A\) has non-zero elements \(1\) in the first row and \(B\) has non-zero elements \(3\) and \(4\) in the second row.

- Define Singular Matrices

Recall that a matrix is singular if its determinant is zero, implying it does not have an inverse.

- Determine if Matrices are Singular

Calculate the determinants:For \(A\):\[|\begin{pmatrix}1 & 0 \ 0 & 0\begin{pmatrix}| = (1)(0) - (0)(0) = 0\]For \(B\):\[|\begin{pmatrix}0 & 0 \ 3 & 4\begin{pmatrix}| = (0)(4) - (0)(3) = 0\]Since both determinants are zero, both matrices are singular.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing a new matrix. The rule for multiplying two matrices, such as A and B, is that the element in the resulting matrix, at position (i, j), is computed by multiplying elements from the i-th row of the first matrix and the j-th column of the second matrix, and then summing these products.
For example, in the matrix multiplication problem given, we have:
\[ \begin{pmatrix}1 & 0\br0 & 0 \end{pmatrix} \begin{pmatrix}0 & 0\br3 & 4 \end{pmatrix} = \begin{pmatrix} (1 \times 0) + (0 \times 3) & (1 \times 0) + (0 \times 4) \br(0 \times 0) + (0 \times 3) & (0 \times 0) + (0 \times 4) \end{pmatrix} = \begin{pmatrix} 0 & 0 \br0 & 0 \end{pmatrix} \]
Notice that each element of the resulting matrix is the sum of products, calculated as outlined above. This explains why the resulting matrix in this exercise is a zero matrix.

Determinant

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, \(\begin{pmatrix}a & b \brc & d \end{pmatrix}\), the determinant is given by the formula: \|A| = a \times d - b \times c\.
Determinants are important because they provide insight into several properties of matrices.
For instance, if the determinant of a matrix is zero, this means the matrix is *singular*, which implies it does not have an inverse.
Let's calculate the determinants of matrices A and B from the exercise:
For matrix A: \|A| = 1 \times 0 - 0 \times 0 = 0\
For matrix B: \|B| = 0 \times 4 - 0 \times 3 = 0\
As both determinants are zero, both matrices are singular.

Singular Matrix

A singular matrix is one that does not have an inverse. This is a key property that can often be determined by checking the matrix's determinant. If the determinant of the matrix is zero, the matrix is classified as singular.
Singular matrices are important in various applications, especially within the field of linear algebra.
In this exercise, both matrices A and B are singular because their determinants are zero, as calculated earlier. Now, it is proven that the product of two singular matrices can result in a zero matrix:
Even without either matrix being zero itself. This understanding is critical because it helps in recognizing that a product of matrices resulting in a zero matrix does not necessarily mean that one of the initial matrices has to be the zero matrix.