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Chapter 6: Problem 58

Explain how to find the multiplicative inverse for a \(2 \times 2\) invertible matrix.

Short Answer

Expert verified
For a given 2x2 matrix A = \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\), if \(ad-bc 鈮� 0\), its inverse A藟鹿 is given by A藟鹿 = 1/(ad-bc) * \(\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\).

Step by step solution

01

Identify the given 2 x 2 matrix

For this exercise, let's consider a 2x2 matrix A given by: A = \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\) where a, b, c, and d are constants.
02

Calculate the determinant of the matrix

The determinant 'det(A)' of the matrix A = \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\) is given by: det(A) = ad - bc. We need to check if det(A) is nonzero because only non-singular matrices (determinant is non-zero) are invertible.
03

Compute the adjoint of the matrix

The adjoint of matrix A, denoted as adj(A), is given by transposing the matrix of the cofactors, that is adj(A) = \(\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\)
04

Calculate the inverse of the matrix

The inverse of A is obtained by dividing the adjoint of the matrix by the determinant 'det(A)', i.e., A藟鹿 = adj(A)/det(A) = 1/(ad-bc) * \(\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\). Therefore, A藟鹿 is the inverse of the matrix A.

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Most popular questions from this chapter

Exercises \(85-87\) will help you prepare for the material covered in the next section. Multiply and write the linear system represented by the following matrix multiplication: $$\left[\begin{array}{lll}a_{1} & b_{1} & c_{1} \\\a_{2} & b_{2} & c_{2} \\\a_{3} & b_{3} & c_{3}\end{array}\right]\left[\begin{array}{l}x \\\y \\\z\end{array}\right]=\left[\begin{array}{l} d_{1} \\\d_{2} \\\d_{3}\end{array}\right]$$

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rrr}-2 & 1 & -1 \\\\-5 & 2 & -1 \\\3 & -1 & 1\end{array}\right]$$

Exercises \(77-79\) will help you prepare for the material covered in the first section of the next chapter. Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}-1\) a. Set \(y-0\) and find the \(x\) -intercepts. b. Set \(x-0\) and find the \(y\) -intercepts.

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{llll}1 & 2 & 0 & 0 \\\0 & 0 & 1 & 0 \\\1 & 3 & 0 & 1 \\\4 & 0 & 0 & 2 \end{array}\right]$$

What is the fastest method for solving a linear system with your graphing utility?
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