Question

Find \(A B\) and \(B A\) given $$ A=\left(\begin{array}{ll} 1 & 2 \\ 3 & 6 \end{array}\right), \quad B=\left(\begin{array}{rr} 10 & 4 \\ -5 & -2 \end{array}\right) $$ Observe that \(A B\) is the null matrix; if we call it 0 , then \(A B=0\), but neither \(A\) nor \(B\) is Show that \(A\) is singular.

Short answer

AB = 0 and BA = \(\begin{pmatrix} 22 & 44 \ -11 & -22 \end{pmatrix}\). A is singular.

Step-by-step solution

Set Up the Matrices

First, define the given matrices: A = \(\begin{pmatrix} 1 & 2 \ 3 & 6 \end{pmatrix}\)B = \(\begin{pmatrix} 10 & 4 \ -5 & -2 \end{pmatrix}\)

Calculate the Product AB

To find the product of matrix \(A\) and matrix \(B\), multiply them using the matrix multiplication rule: \((AB)_{ij} = \sum_{k} A_{ik}B_{kj}\)Calculate each element of the resulting matrix:\((AB)_{11} = 1*10 + 2*(-5) = 10 - 10 = 0\)\((AB)_{12} = 1*4 + 2*(-2) = 4 - 4 = 0\)\((AB)_{21} = 3*10 + 6*(-5) = 30 - 30 = 0\)\((AB)_{22} = 3*4 + 6*(-2) = 12 - 12 = 0\)Thus, the product is:\(AB = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\)

Verify AB is the Zero Matrix

From the calculation, it is observed that every element of matrix \(AB\) is 0. Thus, \(AB = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\). This matrix is called the null matrix.

Calculate the Product BA

To find the product of matrix \(B\) and matrix \(A\), multiply them using the matrix multiplication rule: \((BA)_{ij} = \sum_{k} B_{ik}A_{kj}\)Calculate each element of the resulting matrix:\((BA)_{11} = 10*1 + 4*3 = 10 + 12 = 22\)\((BA)_{12} = 10*2 + 4*6 = 20 + 24 = 44\)\((BA)_{21} = -5*1 + -2*3 = -5 - 6 = -11\)\((BA)_{22} = -5*2 + -2*6 = -10 - 12 = -22\)Thus, the product is:\(BA = \begin{pmatrix} 22 & 44 \ -11 & -22 \end{pmatrix}\)

Check if Matrix A is Singular

A matrix is singular if its determinant is 0. Calculate the determinant of matrix \(A\):\(\text{det}(A) = \begin{vmatrix} 1 & 2 \ 3 & 6 \end{vmatrix} = (1*6) - (2*3) = 6 - 6 = 0\)Since the determinant of \(A\) is 0, \(A\) is singular.

Key concepts

Singular Matrix

A matrix is considered singular if it does not have an inverse. This occurs when the determinant of the matrix is zero.
In the given exercise, matrix \(A\) is defined as:
\[A = \begin{pmatrix} 1 & 2 \ 3 & 6 \end{pmatrix}\]
To determine if \(A\) is singular, calculate its determinant. The determinant of a \(2 \times 2\) matrix \( \begin{pmatrix} a & b \ c & d \ \end{pmatrix} \) is computed as \(ad - bc\).
For matrix \(A\), calculate:
\[ \text{det}(A) = 1*6 - 2*3 = 6 - 6 = 0 \]
Since the determinant is zero, matrix \(A\) is singular. A singular matrix cannot be used in operations that require matrix inversion.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It provides useful information about the matrix, such as whether the matrix is invertible.
For a \(2 \times 2\) matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is found using the formula:
\[ \text{det}(M) = ad - bc \]
Calculating the determinant of matrix \(A\) given in the exercise:
\[ \begin{vmatrix} 1 & 2 \ 3 & 6 \end{vmatrix} = (1*6) - (2*3) = 6 - 6 = 0 \]
A determinant of zero indicates that the matrix is singular, meaning it does not have an inverse.
The determinant is a useful property when solving systems of linear equations, finding eigenvalues, and understanding matrix behavior. It exists only for square matrices.

Null Matrix

A null matrix, also known as a zero matrix, is a matrix in which all the elements are zero.
For example, a \(2 \times 2\) null matrix looks like this:
\[ 0 = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \]
In the exercise, the product of matrices \(A\) and \(B\) resulted in a null matrix:
\[ AB = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \]
This indicates that no matter what multiples and additions occur during the matrix multiplication process, the end result is still zero across all elements.
Null matrices are important in various mathematical contexts, such as solutions to homogeneous systems of linear equations and the analysis of matrix properties.