The value of matrix A is [tex]A =\left[\begin{array}{cc}0&4\\-8&12\end{array}\right][/tex]
How to determine the matrix A?
The given parameters are:
[tex]\lambda_1 = 4[/tex] at [tex]v_1 = \left[\begin{array}{c}1&1\end{array}\right][/tex]
[tex]\lambda_2 = 8[/tex] at [tex]v_2 = \left[\begin{array}{c}1&2\end{array}\right][/tex]
The matrix A is calculated using:
A = S ∧ S⁻¹
Where:
[tex]S = \left[\begin{array}{cc}v_1&v_2\end{array}\right][/tex]
[tex]\lambda = \left[\begin{array}{cc}\lambda_1&0\\0&\lambda_2\end{array}\right][/tex]
[tex]\lambda =\left[\begin{array}{cc}4&0\\0&8\end{array}\right][/tex]
Next, we have:
[tex]S = \left[\begin{array}{cc}v_1&v_2\end{array}\right][/tex]
This gives
[tex]S =\left[\begin{array}{cc}1&1\\1&2\end{array}\right][/tex]
Calculate the determinant of S
|S| = 1 * 2 - 1 * 1
|S| = 1
So, the inverse of S is:
[tex]S^{-1} = \frac{1}{1} * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]\\[/tex]
Evaluate
[tex]S^{-1} = \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]\\[/tex]
Recall that:
A = S ∧ S⁻¹
So, we have:
[tex]A =\left[\begin{array}{cc}1&1\\1&2\end{array}\right] * \left[\begin{array}{cc}4&0\\0&8\end{array}\right] * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right][/tex]
Multiply the first two matrices
[tex]A =\left[\begin{array}{cc}1 * 4 + 1 * 0&1 * 0 + 1 * 8\\1 * 4 + 2 * 0&1 * 0 + 2 * 8\end{array}\right] * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right][/tex]
Evaluate
[tex]A =\left[\begin{array}{cc}4&8\\4&16\end{array}\right] * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right][/tex]
Evaluate the product
[tex]A =\left[\begin{array}{cc}4*2+8*-1&4 * -1 + 8 * 1\\4 * 2 + 16 * -1&4 * -1 + 16 * 1\end{array}\right][/tex]
Evaluate
[tex]A =\left[\begin{array}{cc}0&4\\-8&12\end{array}\right][/tex]
Hence, the value of matrix A is [tex]A =\left[\begin{array}{cc}0&4\\-8&12\end{array}\right][/tex]
Read more about matrices at:
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