Problem 14 - Algebra 2

Question

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{2} & {-1} \\ {1} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

\( \begin{bmatrix} 0 & 1 \ -1 & 2 \end{bmatrix} \)

Step 1: Calculate the Determinant

The determinant of a 2x2 matrix \( A \) with elements \( a, b, c, \text{and} d \) arranged as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). For the given matrix \( \begin{bmatrix} 2 & -1 \ 1 & 0 \end{bmatrix} \), the determinant is \( (2)(0) - (1)(-1) = 0 - (-1) = 1 \) which is non-zero.

Step 2: Find the Inverse using the Adjugate Method

The inverse of a 2x2 matrix \( A \) is \( \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \) where \( \text{det}(A) \) is the determinant of \( A \) from step 1. So, for the given matrix \( \begin{bmatrix} 2 & -1 \ 1 & 0 \end{bmatrix} \), its inverse is \( \frac{1}{1} \begin{bmatrix} 0 & 1 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -1 & 2 \end{bmatrix} \).

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is essential when dealing with linear algebra, especially in finding matrix inverses. In simple terms, the determinant helps us determine if a matrix is invertible and is a special number that can be calculated from its elements. For a 2x2 matrix, the determinant is found by subtracting the product of its diagonals. The formula is expressed as \( ad - bc \) for a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).

If the determinant is zero, the matrix doesn't have an inverse (it is called a singular matrix). However, if the determinant is a non-zero value, the matrix will have an inverse. It's worthwhile to note that determinants can also be found for larger square matrices though the process becomes increasingly complex.

Adjugate Method

Once it's established that a matrix is invertible by its non-zero determinant, the next step is to find its inverse. One popular method to do this is the adjugate method. This involves several steps, starting with finding the matrix of minors, which is turned into the matrix of cofactors by applying a checkerboard pattern of plus and minus signs known as the sign chart.

The adjugate (also called the adjoint) of a matrix is the transpose of this matrix of cofactors. For a 2x2 matrix, this process is simplified since the adjugate is only a matter of swapping the elements on the main diagonal and changing the signs of the other two elements. The final step is multiplying each element of the adjugate by \( 1 / \text{det}(A) \), where \( A \) is the original matrix, and this gives us the inverse.

Inverse of a 2x2 Matrix

Inverting a 2x2 matrix is relatively straightforward especially when compared to matrices of larger sizes. Using the adjugate method mentioned, the inverse of a 2x2 matrix \( A \) can be found with a formula. Assuming \( A \) is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), with \( \text{det}(A) \) being non-zero, the inverse of \( A \) is \( \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).

This method essentially involves flipping the position of \( a \) and \( d \) and then changing the signs of \( b \) and \( c \). After calculating the determinant as covered in the first section, multiply the adjugate matrix by the reciprocal of the determinant. For the matrix in the exercise, the determinant is 1, which simplifies the calculations substantially as multiplying by 1 doesn't change the values.