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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 53
Capitolo 7, Problema 53

In Exercises 53–54, evaluate each determinant. ∣∣31−23∣∣7015∣∣3007∣∣9−635∣∣\(\begin{vmatrix}\) \(\begin{vmatrix}\) 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 \(\end{vmatrix}\) \(\end{vmatrix}\)

Guida verificata passo dopo passo
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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: \(\det(AB) = \det(A) \times \det(B)\).

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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.
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Determinants of 2×2 Matrices

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.
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Determinants of 2×2 Matrices

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.
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Interval Notation
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