https://file.aiquickdraw.com/mathbot/file/1727188461840tdy1o33s.jpg

You have provided a matrix \( A \). To compute \( A \times A^{-1} \), we first need to find the inverse of matrix \( A \). The matrix provided is:

\[
A = \begin{pmatrix}
7 & 2 & 9 & 7 \\
8 & 8 & 9 & 7 \\
15 & -65 & 3 & 13 \\
-3 & -4 & 43 & 35
\end{pmatrix}
\]

The product of a matrix and its inverse is always the identity matrix, so \( A \times A^{-1} \) should give the 4x4 identity matrix:

\[
I = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

Let me calculate the inverse of matrix \( A \) and then perform the multiplication \( A \times A^{-1} \).The determinant of matrix \( A \) is approximately \(-21674\), which confirms that \( A \) is invertible. 

Here is the inverse of matrix \( A \):

\[
A^{-1} = \begin{pmatrix}
0.1433 & -0.0176 & 0.0039 & -0.0266 \\
-0.1906 & 0.1696 & -0.0006 & 0.0044 \\
1.1111 & -0.8524 & -0.0691 & -0.0261 \\
-1.3745 & 1.0651 & 0.0852 & 0.0588
\end{pmatrix}
\]

When multiplying \( A \) by its inverse, we obtain the identity matrix:

\[
A \times A^{-1} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

Let me know if you would like further details or clarifications. Here are some related questions for practice:

1. What properties must a matrix satisfy to have an inverse?
2. Can you calculate the determinant of a 2x2 matrix?
3. How do you find the inverse of a 3x3 matrix?
4. What is the relationship between the determinant and the inverse of a matrix?
5. How does matrix multiplication work for non-square matrices?

**Tip**: Always check if the determinant is non-zero before attempting to find the inverse of a matrix.

Find the inverse of matrix A and compute the product A x A^-1.

Explore how to find the inverse of a matrix A and compute the multiplication of A with its inverse, A x A^-1. This guide covers the calculation of the determinant, the conditions for a matrix to be invertible, and the steps to find the inverse. Suitable for college-level students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation