Question

Either compute the inverse of the given matrix, or else show that it is singular. \(\left(\begin{array}{cc}{3} & {-1} \\ {6} & {2}\end{array}\right)\)

Short answer

Answer: The inverse of the matrix \(\left(\begin{array}{cc}{3} & {-1} \\\ {6} & {2}\end{array}\right)\) is \(\left(\begin{array}{cc}{\frac{1}{6}} & {\frac{1}{12}} \\\ {-\frac{1}{2}} & {\frac{1}{4}}\end{array}\right)\).

Step-by-step solution

Calculate the determinant of the matrix

The determinant of a 2x2 matrix \(\left(\begin{array}{cc}{a} & {b} \\\ {c} & {d}\end{array}\right)\) can be found using the formula \(ad - bc\). So for the given matrix \(\left(\begin{array}{cc}{3} & {-1} \\\ {6} & {2}\end{array}\right)\), the determinant is: \((3)(2)-(-1)(6) = 6+6=12\). Since the determinant is non-zero (12), the matrix is not singular and we can find the inverse.

Find the inverse of the matrix

To find the inverse of a 2x2 matrix \(\left(\begin{array}{cc}{a} & {b} \\\ {c} & {d}\end{array}\right)\), first find the adjugate matrix which is \(\left(\begin{array}{cc}{d} & {-b} \\\ {-c} & {a}\end{array}\right)\) and then multiply this matrix by \(\frac{1}{\text{determinant}}\). Thus, the adjugate matrix is: \(\left(\begin{array}{cc}{2} & {1} \\\ {-6} & {3}\end{array}\right)\). Now, multiply the adjugate matrix by \(\frac{1}{12}\) to get the inverse of the matrix: \((\frac{1}{12})\left(\begin{array}{cc}{2} & {1} \\\ {-6} & {3}\end{array}\right) = \left(\begin{array}{cc}{\frac{1}{6}} & {\frac{1}{12}} \\\ {-\frac{1}{2}} & {\frac{1}{4}}\end{array}\right)\). So the inverse of the matrix \(\left(\begin{array}{cc}{3} & {-1} \\\ {6} & {2}\end{array}\right)\) is: \(\left(\begin{array}{cc}{\frac{1}{6}} & {\frac{1}{12}} \\\ {-\frac{1}{2}} & {\frac{1}{4}}\end{array}\right)\).

Key concepts

Determinant Calculation

Understanding the determinant of a matrix is crucial in linear algebra, especially when discussing matrix inversion. For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is a special number calculated from its elements using the formula \(ad - bc\).
This number helps us determine whether a matrix is invertible or singular.

• If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
• If the determinant is not zero, the matrix is non-singular, meaning it has an inverse.

In the given exercise, the determinant of the matrix \(\begin{pmatrix} 3 & -1 \ 6 & 2 \end{pmatrix}\) is calculated as follows:- Multiply the first diagonal: \(3 \times 2 = 6\).- Multiply the second diagonal: \(-1 \times 6 = -6\).- Subtract the second result from the first: \(6 - (-6) = 12\).With a determinant of \(12\), we confirm that the matrix is non-singular and can be inverted.

Adjugate Matrix

The adjugate matrix is an essential component in determining the inverse of a matrix. For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), calculating the adjugate involves rearranging and changing the signs of certain elements.
Here’s how you find the adjugate:

• Swap the positions of \(a\) and \(d\).
• Change the signs of \(b\) and \(c\).

So, the adjugate of \(\begin{pmatrix} 3 & -1 \ 6 & 2 \end{pmatrix}\) is \(\begin{pmatrix} 2 & 1 \ -6 & 3 \end{pmatrix}\).This transformation helps prepare the matrix for the multiplication step needed to compute the inverse. It's like reshuffling the pieces to get ready for the final calculation.

2x2 Matrix Inverse

Finding the inverse of a 2x2 matrix is made easy with a step-by-step approach. After obtaining the adjugate matrix, the next step is to use the determinant calculated earlier.
The formula for the inverse of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is:\[\frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\]In this exercise, the inverse of the matrix \(\begin{pmatrix} 3 & -1 \ 6 & 2 \end{pmatrix}\) involves these steps:

• Use the determinant (12) for scaling.
• Apply the adjugate matrix \(\begin{pmatrix} 2 & 1 \ -6 & 3 \end{pmatrix}\).
• Multiply each element of the adjugate by \(\frac{1}{12}\).

This calculation results in an inverse \(\begin{pmatrix} \frac{1}{6} & \frac{1}{12} \ -\frac{1}{2} & \frac{1}{4} \end{pmatrix}\). Using the inverse, you can now solve systems of linear equations where the matrix is a coefficient matrix, just like a professional mathematician.