Question

Given \(A=\left(\begin{array}{rr}5 & -3 \\ -2 & 1\end{array}\right)\), then \(\mathrm{A}^{-1}=\) _____. (1) \(\left(\begin{array}{rr}-1 & -3 \\ 2 & 5\end{array}\right)\) (2) \(\left(\begin{array}{rr}-1 & -3 \\ -2 & -5\end{array}\right)\) (3) \(\left(\begin{array}{rr}-1 & -3 \\ 2 & -5\end{array}\right)\) (4) \(\left(\begin{array}{rr}1 & 3 \\ 2 & -5\end{array}\right)\)

Short answer

Answer: \(A^{-1}\) = \(\left( \begin{array}{rr} -1 & -3 \\\ -2 & -5 \end{array} \right)\)

Step-by-step solution

Calculate the determinant of A

The determinant of A is given by the formula ad - bc, where A = \(\left( \begin{array}{rr} a & b \\\ c & d \end{array} \right)\). In our case, A = \(\left( \begin{array}{rr} 5 & -3 \\\ -2 & 1\end{array}\right)\). So a = 5, b = -3, c = -2, and d = 1. Det(A) = ad - bc = (5)(1) - (-3)(-2) = 5 - 6 = -1.

Calculate the inverse of A

With det(A) = -1, we can now calculate the inverse of A using the formula: \(A^{-1} = \frac{1}{\mathrm{det}(A)} \cdot \left( \begin{array}{rr} d & -b \\\ -c & a \end{array} \right)\) \(A^{-1} = \frac{1}{-1} \cdot \left( \begin{array}{rr} 1 & 3 \\\ 2 & 5 \end{array} \right)\) \(A^{-1} = \left( \begin{array}{rr} -1 & -3 \\\ -2 & -5 \end{array} \right)\)

Compare A^-1 with answer choices

Comparing the calculated A^-1 with the given answer options in the problem, we see that \(A^{-1}\) = \(\left( \begin{array}{rr} -1 & -3 \\\ -2 & -5 \end{array} \right)\) matches option (2): \(\left(\begin{array}{rr}-1 & -3 \\\ -2 & -5\end{array}\right)\). Hence, the correct inverse of the given matrix A is option (2) \(\left(\begin{array}{rr}-1 & -3 \\\ -2 & -5\end{array}\right)\).

Key concepts

Determinant of a Matrix

Understanding the determinant of a matrix is crucial when dealing with inverse matrices. The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, such as \( A = \left( \begin{array}{rr} a & b \ c & d \end{array} \right) \), the determinant is calculated using the formula \( \text{det}(A) = ad - bc \). This value tells us important information about the matrix, such as whether or not the matrix is invertible.

In the context of matrix inversion, a matrix is invertible only if its determinant is not zero. If the determinant is zero, the matrix does not have an inverse, meaning we cannot find a matrix that, when multiplied by the original, results in the identity matrix. In our example, the matrix is given by \( A = \left( \begin{array}{rr} 5 & -3 \ -2 & 1 \end{array} \right) \).

By substituting the corresponding values into the formula, we find the determinant as follows:

• \( a = 5 \), \( b = -3 \), \( c = -2 \), and \( d = 1 \)
• \( \text{det}(A) = (5 \cdot 1) - (-3 \cdot -2) = 5 - 6 = -1 \)

This negative value indicates that the matrix is invertible.

2x2 Matrix

A 2x2 matrix is a simple yet foundational element of linear algebra. It consists of two rows and two columns, forming a grid of four elements. Typically represented as:

\[\begin{array}{rr}a & b \c & d\end{array}\]

Each position in the matrix has a distinct role. The elements \( a \), \( b \), \( c \), and \( d \) are numbers that determine the characteristics and properties of the matrix when used in calculations, such as finding a determinant or inverse.

2x2 matrices are often used in various fields, including physics, computer graphics, and engineering, where they help solve systems of linear equations and transform vectors. They might appear small, but their applications and importance are vast. Knowing how to manage these matrices, especially when calculating their determinants and inverses, is essential for students and professionals who work with matrices.

Inverse Matrix Formula

Calculating the inverse of a 2x2 matrix involves a specific formula that utilizes the matrix's determinant. The inverse matrix \( A^{-1} \) of a given 2x2 matrix \( A = \left( \begin{array}{rr} a & b \ c & d \end{array} \right) \) is found using:

\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \left( \begin{array}{rr} d & -b \ -c & a \end{array} \right) \]

Note that this formula is valid only if the determinant \( \text{det}(A) \) is not zero. Inverting a matrix essentially means finding a matrix that, when multiplied by the original, results in the identity matrix \( I \).

Using our example matrix \( A = \left( \begin{array}{rr} 5 & -3 \ -2 & 1 \end{array} \right) \), we already calculated \( \text{det}(A) = -1 \). Thus, the inverse is:

• Plug in values into the formula: \( A^{-1} = \frac{1}{-1} \cdot \left( \begin{array}{rr} 1 & 3 \ 2 & 5 \end{array} \right) \)
• Simplify the expression: \( A^{-1} = \left( \begin{array}{rr} -1 & -3 \ -2 & -5 \end{array} \right) \)

This matrix is the inverse of our original matrix, confirming that when this inverse is multiplied by \( A \), it yields the identity matrix, demonstrating its fundamental property.