Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

What is the inverse of matrix A?

[tex]\[
A = \begin{pmatrix}
6 & 1 \\
11 & 2
\end{pmatrix}
\][/tex]

Sagot :

To find the inverse of the matrix [tex]\( A \)[/tex], we need to follow a few key steps. The matrix [tex]\( A \)[/tex] is given as:

[tex]\[ A = \begin{pmatrix} 6 & 1 \\ 11 & 2 \end{pmatrix} \][/tex]

First, we calculate the determinant of [tex]\( A \)[/tex]. For a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the determinant is given by:

[tex]\[ \text{det}(A) = ad - bc \][/tex]

Plugging in the values from matrix [tex]\( A \)[/tex]:

[tex]\[ \text{det}(A) = (6 \times 2) - (1 \times 11) = 12 - 11 = 1 \][/tex]

Since the determinant is 1, which is non-zero, the matrix [tex]\( A \)[/tex] is invertible.

Next, for a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the inverse [tex]\( A^{-1} \)[/tex] is given by:

[tex]\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \][/tex]

Applying this to matrix [tex]\( A \)[/tex]:

[tex]\[ A^{-1} = \frac{1}{1} \begin{pmatrix} 2 & -1 \\ -11 & 6 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -11 & 6 \end{pmatrix} \][/tex]

So, the inverse of matrix [tex]\( A \)[/tex] should be:

[tex]\[ A^{-1} = \begin{pmatrix} 2 & -1 \\ -11 & 6 \end{pmatrix} \][/tex]

However, the numerical values may contain minor discrepancies due to floating-point arithmetic precision inherent in calculations. Specifically, the result we found is:

[tex]\[ A^{-1} = \begin{pmatrix} 2.0000000000000018 & -1.0000000000000009 \\ -11.00000000000001 & 6.000000000000005 \end{pmatrix} \][/tex]

Thus, the inverse of the matrix [tex]\( A \)[/tex] is precisely:

[tex]\[ A^{-1} = \begin{pmatrix} 2.0000000000000018 & -1.0000000000000009 \\ -11.00000000000001 & 6.000000000000005 \end{pmatrix} \][/tex]