To find the inverse of matrix [tex]\( B \)[/tex], we need to determine [tex]\( B^{-1} \)[/tex] such that [tex]\( B \cdot B^{-1} = I \)[/tex], where [tex]\( I \)[/tex] is the identity matrix. To solve for [tex]\( B^{-1} \)[/tex], we use methods such as Gaussian elimination, the adjoint method, or matrix algebra properties.
Given matrix [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix}
3 & 4 & -1 \\
1 & 0 & 3 \\
2 & 5 & -4
\end{pmatrix}
\][/tex]
The inverse [tex]\( B^{-1} \)[/tex] is:
[tex]\[
B^{-1} = \begin{pmatrix}
1.5 & -1.1 & -1.2 \\
-1 & 1 & 1 \\
-0.5 & 0.7 & 0.4
\end{pmatrix}
\][/tex]
This matrix, when multiplied by [tex]\( B \)[/tex], gives the identity matrix. This solution matrix can be verified through matrix multiplication, ensuring that:
[tex]\[
B \cdot B^{-1} = I,
\][/tex]
where [tex]\( I \)[/tex] is the [tex]\( 3 \times 3 \)[/tex] identity matrix:
[tex]\[
I = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\][/tex]
Thus, the inverse of matrix [tex]\( B \)[/tex] is:
[tex]\[
B^{-1} = \begin{pmatrix}
1.5 & -1.1 & -1.2 \\
-1 & 1 & 1 \\
-0.5 & 0.7 & 0.4
\end{pmatrix}
\][/tex]
This result is derived by appropriately solving the matrix equations and verifying the correctness of [tex]\( B^{-1} \)[/tex].
