Find the determinant of M = [[3, −5, −5], [−6, −3, −3], [2, 2, 1]]. If P = [[8, 3], [−9, 2]], find P inverse.

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Question 2: Find the determinant of $M = \begin{bmatrix} 3 & -5 & -5 \\ -6 & -3 & 2 \\ 2 & 2 & 1 \end{bmatrix}$ .

Write the formula for the determinant of a $3 \times 3$ matrix.
For a matrix $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ , the determinant is $a(ei - fh) - b(di - fg) + c(dh - eg)$ .
Substitute the values from matrix $M$ into the formula.
$\det(M) = 3((-3)(1) - (2)(2)) - (-5)((-6)(1) - (2)(2)) + (-5)((-6)(2) - (-3)(2))$
Calculate the terms inside the parentheses.
$\det(M) = 3(-3 - 4) + 5(-6 - 4) - 5(-12 - (-6))$ $\det(M) = 3(-7) + 5(-10) - 5(-12 + 6)$ $\det(M) = 3(-7) + 5(-10) - 5(-6)$
Perform the multiplications and additions.
$\det(M) = -21 - 50 + 30$ $\det(M) = -71 + 30$ $\det(M) = -41$
The determinant of $M$ is $\boxed{-41}$ .

Question 1b: If $P = \begin{bmatrix} 8 & 3 \\ -9 & 2 \end{bmatrix}$ , find $P^{-1}$ (the inverse of P).
Calculate the determinant of matrix $P$ .
For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is $ad - bc$ .
$\det(P) = (8)(2) - (3)(-9)$ $\det(P) = 16 - (-27)$ $\det(P) = 16 + 27$ $\det(P) = 43$
Write the formula for the inverse of a $2 \times 2$ matrix.
$P^{-1} = \frac{1}{\det(P)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
Substitute the values into the inverse formula.
$P^{-1} = \frac{1}{43} \begin{bmatrix} 2 & -3 \\ -(-9) & 8 \end{bmatrix}$ $P^{-1} = \frac{1}{43} \begin{bmatrix} 2 & -3 \\ 9 & 8 \end{bmatrix}$
Multiply each element by $\frac{1}{43}$ .
$P^{-1} = \begin{bmatrix} \frac{2}{43} & \frac{-3}{43} \\ \frac{9}{43} & \frac{8}{43} \end{bmatrix}$
The inverse of $P$ is $\boxed{\begin{bmatrix} \frac{2}{43} & \frac{-3}{43} \\ \frac{9}{43} & \frac{8}{43} \end{bmatrix}}$ .

Find the determinant of M = [[3, -5, -5], [-6, -3, -3], [2, 2, 1]]. If P = [[8, 3], [-9, 2]], find P inverse.

Step-by-step solution

Write the formula for the determinant of a $3 \times 3$ matrix.

Substitute the values from matrix $M$ into the formula.

Calculate the terms inside the parentheses.

Perform the multiplications and additions.

Calculate the determinant of matrix $P$ .

Write the formula for the inverse of a $2 \times 2$ matrix.

Substitute the values into the inverse formula.

Multiply each element by $\frac{1}{43}$ .

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