Question

Only one of the following matrices has an inverse. Find the determinant of each matrix, and use the determinants to identify the one that has an inverse. Then find the inverse. A= [1 4 1 ] [0 2 0 ] [1 0 1 ] , B=[1 4 0 ] [0 2 0 ] [-3 0 1 ]

Answer 1

Both matrices A and B have non-zero determinants of 2, meaning they both technically have inverses.Assuming matrix A was intended to be the one with an inverse, its inverse can be found using a method that involves its adjugate and determinant.

Step-by-step explanation:

In mathematics, particularly in linear algebra, the inverse of a matrix exists only if the determinant of the matrix is not zero. The determinant is a special value that can be calculated from the elements of a square matrix. We will calculate the determinants of matrices A and B to determine which one has an inverse.

For matrix A:

A =
\begin{bmatrix}
1 & 4 & 1 \\
0 & 2 & 0 \\
1 & 0 & 1
\end{bmatrix}

The determinant of matrix A can be calculated as follows:

det(A) = 1(2*1 - 0*0) - 4(0*1 - 0*1) + 1(0*0 - 2*1) = 2.

Since the determinant of matrix A is non-zero, it has an inverse.

For matrix B:

B =
\begin{bmatrix}
1 & 4 & 0 \\
0 & 2 & 0 \\
-3 & 0 & 1
\end{bmatrix}

The determinant of matrix B is:

det(B) = 1(2*1 - 0*0) - 4(0*1 - 0*(-3)) + 0(0*0 - 2*(-3)) = 2 - 0 + 0 = 2.

Both matrices A and B have non-zero determinants, which means they both have inverses.