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=$$I2A^{-1} A$$ = $$I_2.A$$=[3−1−42]A = \begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}A=[3−4−12]

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 15

7장, 문제 15

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 = $\begin{bmatrix}$ 3 & -1 \\ -4 & 2 $\end{bmatrix}$$

검증된 단계별 안내

Identify the elements of matrix $A$: $a = 3$, $b = -1$, $c = -4$, and $d = 2$.

Calculate the determinant of $A$ using the formula $ad - bc$: compute $3 \times 2 - (-1) \times (-4)$.

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

Multiply the scalar $\frac{1}{ad - bc}$ by the matrix $\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ to write the inverse matrix explicitly.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Matrix Inverse

The inverse of a square matrix A, denoted A⁻¹, is a matrix that when multiplied by A yields the identity matrix I. For a 2x2 matrix, the inverse exists only if the determinant (ad - bc) is nonzero. The formula for the inverse involves swapping elements a and d, changing signs of b and c, and dividing by the determinant.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix A = [[a, b], [c, d]] is calculated as ad - bc. It is a scalar value that indicates whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides information about the matrix's scaling factor and orientation.

Identity Matrix and Matrix Multiplication

The identity matrix I₂ is a 2x2 matrix with ones on the diagonal and zeros elsewhere. Multiplying any 2x2 matrix A by I₂ leaves A unchanged. To verify an inverse, multiply A by A⁻¹ and A⁻¹ by A; both products should equal I₂, confirming the correctness of the inverse.