Question

I really need help with this matrix question. Please and thank you!.

Answer 1

The value of matrix A is $A =\left[\begin{array}{cc}0&4\\-8&12\end{array}\right]$

How to determine the matrix A?

The given parameters are:

$\lambda_1 = 4$ at $v_1 = \left[\begin{array}{c}1&1\end{array}\right]$

$\lambda_2 = 8$ at $v_2 = \left[\begin{array}{c}1&2\end{array}\right]$

The matrix A is calculated using:

A = S ∧ S⁻¹

Where:

$S = \left[\begin{array}{cc}v_1&v_2\end{array}\right]$

$\lambda = \left[\begin{array}{cc}\lambda_1&0\\0&\lambda_2\end{array}\right]$

$\lambda =\left[\begin{array}{cc}4&0\\0&8\end{array}\right]$

Next, we have:

$S = \left[\begin{array}{cc}v_1&v_2\end{array}\right]$

This gives

$S =\left[\begin{array}{cc}1&1\\1&2\end{array}\right]$

Calculate the determinant of S

|S| = 1 * 2 - 1 * 1

|S| = 1

So, the inverse of S is:

$S^{-1} = \frac{1}{1} * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]$

Evaluate

$S^{-1} = \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]$

Recall that:

A = S ∧ S⁻¹

So, we have:

$A =\left[\begin{array}{cc}1&1\\1&2\end{array}\right] * \left[\begin{array}{cc}4&0\\0&8\end{array}\right] * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]$

Multiply the first two matrices

$A =\left[\begin{array}{cc}1 * 4 + 1 * 0&1 * 0 + 1 * 8\\1 * 4 + 2 * 0&1 * 0 + 2 * 8\end{array}\right] * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]$

Evaluate

$A =\left[\begin{array}{cc}4&8\\4&16\end{array}\right] * \left[\begin{array}{cc}2&-1\\-1&1\end{array}\right]$

Evaluate the product

$A =\left[\begin{array}{cc}4*2+8*-1&4 * -1 + 8 * 1\\4 * 2 + 16 * -1&4 * -1 + 16 * 1\end{array}\right]$

Evaluate

$A =\left[\begin{array}{cc}0&4\\-8&12\end{array}\right]$

Hence, the value of matrix A is $A =\left[\begin{array}{cc}0&4\\-8&12\end{array}\right]$

Read more about matrices at:

brainly.com/question/11989522

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