In Exercises 53–54, evaluate each determinant. $\begin{vmatrix}\) $\begin{vmatrix}$ 3 & 1 \\ -2 & 3 $\end{vmatrix}$ & $\begin{vmatrix}$ 7 & 0 \\ 1 & 5 $\end{vmatrix}$ \\ $\begin{vmatrix}$ 3 & 0 \\ 0 & 7 $\end{vmatrix}$ & $\begin{vmatrix}$ 9 & -6 \\ 3 & 5 $\end{vmatrix}$ \(\end{vmatrix}$

검증된 단계별 안내

Step 1: Recognize that the problem asks to evaluate the determinant of the product of two matrices. The determinant of a product of matrices equals the product of their determinants. So, we use the property: $\det(AB) = \det(A) \times \det(B)$.

Step 2: Identify the two matrices from the problem. Let matrix $A = \begin{bmatrix} 3 & 1 \\ -2 & 3 \end{bmatrix}$ and matrix $B = \begin{bmatrix} 7 & 0 \\ 1 & 5 \end{bmatrix}$.

Step 3: Calculate the determinant of matrix $A$ using the formula for a 2x2 matrix: $\det(A) = a d - b c$, where $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. So, $\det(A) = (3)(3) - (1)(-2)$.

Step 4: Calculate the determinant of matrix $B$ similarly: $\det(B) = (7)(5) - (0)(1)$.

Step 5: Multiply the two determinants found in steps 3 and 4 to get the determinant of the product matrix: $\det(AB) = \det(A) \times \det(B)$.

비슷한 문제에 대한 검증된 영상 답변:

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주요 개념

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

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine properties like invertibility and area scaling in linear transformations. Understanding this formula is essential for evaluating the determinants in the given problem.

Properties of Determinants

Determinants have properties such as linearity, the effect of row operations, and multiplicative behavior. For example, the determinant of a product of matrices equals the product of their determinants. Recognizing these properties can simplify calculations and verify results.

Matrix Notation and Evaluation

Interpreting matrix notation correctly is crucial for evaluating determinants. Each matrix element must be identified accurately, and the determinant formula applied carefully. This ensures correct computation, especially when dealing with multiple matrices as in the exercise.