Question

Find the inverse of each matrix. $$\left[\begin{array}{ll}5 & 3 \\4 & 2\end{array}\right]$$

   Find the inverse of each matrix.
$$\left[\begin{array}{ll}5 & 3 \\4 & 2\end{array}\right]$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 8, Problem 12

Instant Answer

Solved by Expert Joshua Sieverding

06/15/2020

Step 1

The determinant of a 2x2 matrix is calculated as follows: $$\text{det}(A) = ad - bc$$ where a, b, c, and d are the elements of the matrix. In this case, a = 5, b = 3, c = 4, and d = 2. So, the determinant is: $$\text{det}(A) = 5*2 - 3*4 = 10 - 12 = -2$$ Show more…

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Find the inverse of each matrix. $$\left[\begin{array}{ll}5 & 3 \\4 & 2\end{array}\right]$$

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Key Concepts

Matrix Inverse

The matrix inverse of a square matrix is a matrix that, when multiplied by the original matrix, yields the identity matrix. It is analogous to the reciprocal of a number in regular arithmetic. For a matrix to have an inverse, it must be non-singular (i.e., its determinant is not zero). This concept is fundamental in solving systems of linear equations, among other applications in linear algebra.

Determinant

The determinant is a scalar value computed from the elements of a square matrix that provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix, the determinant is calculated as (ad - bc) when the matrix is structured as [a, b; c, d]. A non-zero determinant indicates that the inverse exists, making the determinant a crucial concept in inverse matrix computation.

2x2 Matrix Inverse Formula

For a 2x2 matrix, there is a direct formula to find its inverse. If the matrix is given by [a, b; c, d], the inverse is calculated as 1/(ad - bc) multiplied by the matrix [d, -b; -c, a]. This formula leverages the concept of the determinant and shows how the positions and signs of the elements must be adjusted to obtain the inverse, making it a practical tool for quickly inverting 2x2 matrices.

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