Question

Each matrix is nonsingular. Find the inverse of each matrix. $$ \left[\begin{array}{ll} 2 & 1 \\ a & a \end{array}\right] \quad a \neq 0 $$

Short answer

\[ \begin{bmatrix} 1 & -\frac{1}{a} \ -1 & \frac{2}{a} \end{bmatrix} \]

Step-by-step solution

- Determine the formula for the inverse of a 2x2 Matrix

The inverse of a 2x2 matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] is given by the formula \[ \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \].

- Identify the elements of the given matrix

In the given matrix \[ \begin{bmatrix} 2 & 1 \ a & a \end{bmatrix} \], the elements are: a = 2, b = 1, c = a, d = a.

- Compute the determinant

The determinant of the given matrix is calculated by \[ ad - bc \], so \[ (2)(a) - (1)(a) = 2a - a = a \].

- Apply the inverse formula

Using the determinant and substituting the values back into the inverse formula: \[ \frac{1}{a} \begin{bmatrix} a & -1 \ -a & 2 \end{bmatrix} \].

- Simplify the inverse matrix

Distribute \[ \frac{1}{a} \] through the matrix: \[ \begin{bmatrix} 1 & -\frac{1}{a} \ -1 & \frac{2}{a} \end{bmatrix} \].

Key concepts

Matrix Determinant

The determinant of a matrix is a special number that can tell us a lot about the matrix. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula: \(ad - bc\). This value is crucial because it helps us determine if a matrix has an inverse. If the determinant is zero, the matrix is nonsingular, meaning it does not have an inverse. However, if the determinant is not zero, the matrix is invertible. In our given problem, the matrix is \(\begin{bmatrix} 2 & 1 \ a & a \end{bmatrix}\). Using the formula \(ad - bc\), we find the determinant to be \(2a - a = a\), which tells us it is invertible since \(a eq 0\).

Matrix Inversion

Matrix inversion is the process of finding the inverse of a matrix. For those learning about this concept, it is essential to understand that only certain matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant. The formula for the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is: \(\frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). This formula is directly related to the determinant. In our exercise, we substitute the given values to arrive at: \(\frac{1}{a} \begin{bmatrix} a & -1 \ -a & 2 \end{bmatrix}\). Finally, we multiply through by \(\frac{1}{a}\) to get the simplified inverse: \(\begin{bmatrix} 1 & -\frac{1}{a} \ -1 & \frac{2}{a} \end{bmatrix}\).

2x2 Matrix

Understanding 2x2 matrices is foundational in linear algebra. These small matrices are usually one of the first types students encounter. A 2x2 matrix has two rows and two columns and looks like this: \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). They are particularly simple compared to larger matrices and provide a gentle introduction to more complex matrix operations. With 2x2 matrices, you can easily compute determinants, find inverses, and perform other operations manually. In our exercise, we worked with the matrix \(\begin{bmatrix} 2 & 1 \ a & a \end{bmatrix}\), showcasing typical manipulations like determinant calculation and matrix inversion. Remember, having a solid grasp of these operations for 2x2 matrices sets the stage for understanding larger, more complex matrices.