Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$Consider this matrix: [[-4,2],[-6,6]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{ll} -4 & 2 \\ -6 & 6 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

View step-by-step help

Full solution

Q. Consider this matrix:

\newline

\left[\begin{array}{ll} -4 & 2 \\ -6 & 6 \end{array}\right]

\newline

Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Write Matrix Determinant: Write down the matrix and its determinant. $\newline$ The determinant of a $2$ x $2$ matrix $\left[\begin{array}{cc}a & b \ c & d\end{array}\right]$ is $ad - bc$ . $\newline$ For the matrix $\left[\begin{array}{cc}-4 & 2 \ -6 & 6\end{array}\right]$ , the determinant is $(-4)(6) - (2)(-6)$ .
Calculate Determinant: Calculate the determinant. $\newline$ Determinant = $(-4)(6) - (2)(-6) = -24 - (-12) = -24 + 12 = -12$ .
Check Non-Zero Determinant: Check if the determinant is non-zero. $\newline$ Since the determinant is $-12$ , which is non-zero, the matrix has an inverse.
Inverse Matrix Formula: Write down the formula for the inverse of a $2 \times 2$ matrix. $\newline$ The inverse of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is $\frac{1}{\text{determinant}} \times \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ .
Apply Formula for Inverse: Apply the formula to find the inverse matrix. Using the determinant from Step $2$ and the values from the original matrix, the inverse matrix is $\frac{1}{(-12)}$ * $\left[\begin{array}{cc} 6 & -2 \ 6 & -4 \end{array}\right]$ .
Multiply by Scalar: Multiply each element of the matrix by the scalar $\frac{1}{(-12)}$ . Inverse matrix = $\left[\begin{array}{cc} \left(\frac{1}{(-12)}\right)\cdot 6 & \left(\frac{1}{(-12)}\right)\cdot (-2) \ \left(\frac{1}{(-12)}\right)\cdot 6 & \left(\frac{1}{(-12)}\right)\cdot (-4) \end{array}\right]$ .
Simplify Inverse Matrix: Simplify the matrix. $\newline$ Inverse matrix = \left[\begin{array}{cc}$\newline-\frac{1}{2} & \frac{1}{6} (\newline$-\frac{1}{2} & \frac{1}{3} $\newline$ \end{array}\right]\).
Check Result with Identity: Check the result by multiplying the original matrix by the inverse matrix to see if it yields the identity matrix. Multiplying the original matrix by the inverse should give us the identity matrix if the inverse is correct.