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Math  /  Algebra

QuestionPart 1 Find the determinant of the following matrix: [818−32]\left[\begin{array}{cc} 8 & 18 \\ -3 & 2 \end{array}\right]
70 - Part 2
Does this matrix have an inverse? ✓\checkmark A. Yes B. No
Part 3 Find the inverse of the matrix. □\square
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Studdy Solution

STEP 1

1. The matrix is a 2x2 matrix.
2. The determinant of a 2x2 matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} is calculated as ad−bcad - bc.
3. A matrix has an inverse if and only if its determinant is non-zero.
4. The inverse of a 2x2 matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} is given by 1ad−bc[d−b−ca]\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}, provided the determinant is non-zero.

STEP 2

1. Calculate the determinant of the matrix.
2. Determine if the matrix has an inverse.
3. Find the inverse of the matrix if it exists.

STEP 3

Calculate the determinant of the matrix [818−32]\begin{bmatrix} 8 & 18 \\ -3 & 2 \end{bmatrix}.
The determinant is given by:
det=(8)(2)−(18)(−3) \text{det} = (8)(2) - (18)(-3)
det=16+54 \text{det} = 16 + 54
det=70 \text{det} = 70

STEP 4

Determine if the matrix has an inverse.
Since the determinant is 7070, which is non-zero, the matrix does have an inverse.
Answer: ✓\checkmark A. Yes

STEP 5

Find the inverse of the matrix [818−32]\begin{bmatrix} 8 & 18 \\ -3 & 2 \end{bmatrix}.
The inverse is given by:
170[2−1838] \frac{1}{70} \begin{bmatrix} 2 & -18 \\ 3 & 8 \end{bmatrix}
=[270−1870370870] = \begin{bmatrix} \frac{2}{70} & \frac{-18}{70} \\ \frac{3}{70} & \frac{8}{70} \end{bmatrix}
Simplify the fractions:
=[135−935370435] = \begin{bmatrix} \frac{1}{35} & \frac{-9}{35} \\ \frac{3}{70} & \frac{4}{35} \end{bmatrix}
The inverse of the matrix is:
[135−935370435] \begin{bmatrix} \frac{1}{35} & \frac{-9}{35} \\ \frac{3}{70} & \frac{4}{35} \end{bmatrix}

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