Use the fact that if A=[abcd]A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}, then A−1=1ad−bc[d−b−ca]$$A^{-1}$$=\frac{1}{ad-bc}\begin{bmatrix}d & -b\\ -c & a\end{bmatrix} to find the inverse of each matrix, if possible. Check that AA−1=$$I2AA^{-1}$$ = $$I_2$$ and A−1A=$$I2A^{-1}A$$ = $$I_2.A$$=[10−2−51]A = \begin{bmatrix} 10 & -2 \\ -5 & 1 \end{bmatrix}A=[10−5−21]

Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 17

Chapter 7, Problem 17

Use the fact that if $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ to find the inverse of each matrix, if possible. Check that $A A^{- 1} = I_{2}$ and $A^{- 1} A = I_{2}$ .
$A = $\begin{bmatrix}$10 & -2 \\-5 & 1$\end{bmatrix}$$

Verified step by step guidance

Identify the elements of the matrix $A$ as $a = 10$, $b = -2$, $c = -5$, and $d = 1$.

Calculate the determinant of $A$ using the formula $ad - bc$, which means compute $10 \times 1 - (-2) \times (-5)$.

Check if the determinant is non-zero. If it is zero, the inverse does not exist; if non-zero, proceed to find the inverse.

Use the formula for the inverse matrix: $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$, substituting the values of $a$, $b$, $c$, and $d$.

Verify the result by multiplying $A$ and $A^{-1}$ in both orders ($AA^{-1}$ and $A^{-1}A$) to check if the product is the identity matrix $I_2$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Inverse Formula for 2x2 Matrices

The inverse of a 2x2 matrix A = [[a, b], [c, d]] exists if and only if the determinant (ad - bc) is nonzero. The inverse is given by A^(-1) = (1/(ad - bc)) * [[d, -b], [-c, a]]. This formula allows us to find the inverse by swapping and negating elements and scaling by the reciprocal of the determinant.

Determinant and Its Role in Invertibility

The determinant of a 2x2 matrix, calculated as ad - bc, determines whether the matrix is invertible. If the determinant is zero, the matrix has no inverse. A nonzero determinant ensures the matrix is invertible and the inverse can be computed using the formula.

Verification of Matrix Inverse Using Identity Matrix

To confirm that a matrix B is the inverse of A, multiply A by B and B by A. Both products should yield the 2x2 identity matrix I_2 = [[1, 0], [0, 1]]. This verification step ensures the correctness of the computed inverse matrix.

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