Evaluate each determinant in Exercises 1–10. ∣ − 7 14 2 − 4 ∣ \begin{vmatrix}-7 & 14 \\2 & -4\end{vmatrix} − 7 2 14 − 4

Welcome back. I am so glad you're here. We are given this two by two. Matrix. The first column is negative 45 and the second column is 12 negative six. And we are asked to evaluate the second order determinant. Will we recall from previous lessons on second order determinants that we take the term in the upper left hand corner and multiply it by the term in the bottom right hand corner. So negative four times negative six which is a positive 24. Next we take the term in the bottom left hand corner and multiply it by the term in the upper right hand corner five times 12 is 60. And then we are going to subtract these two terms. So the first term 24 minus the second term 60 Gives us a negative 36 and that is our determinant. We look at our answer choices in this matches with answer choice C. Well done. We'll catch you on the next one.

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

1:05 minutes

Problem 5

Textbook Question

Evaluate each determinant in Exercises 1–10.
$\begin{vmatrix}-7 & 14 \\2 & -4\end{vmatrix}$

Verified step by step guidance

Identify the given 2x2 matrix as $ \begin{bmatrix} -7 & 14 \\ 2 & -4 \end{bmatrix} $.

Recall the formula for the determinant of a 2x2 matrix $ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ is given by $ \text{det} = ad - bc $.

Substitute the values from the matrix into the formula: $ a = -7, b = 14, c = 2, d = -4 $.

Calculate the product of the diagonal elements: $ a \times d = (-7) \times (-4) $.

Calculate the product of the off-diagonal elements: $ b \times c = 14 \times 2 $, then subtract this from the first product to find the determinant.

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Key Concepts

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

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.

Matrix Notation and Elements

Understanding matrix notation involves recognizing the position of elements: 'a' and 'b' in the first row, 'c' and 'd' in the second. Correctly identifying these values is essential for accurate determinant calculation.

Application of Determinants in Algebra

Determinants are used to solve systems of linear equations, find inverses of matrices, and analyze linear transformations. Evaluating determinants is a fundamental skill in college algebra for these applications.

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