Evaluate each determinant in Exercises 1–10. ∣ − 5 − 1 − 2 − 7 ∣ \begin{vmatrix}-5 & -1 \\-2 & -7\end{vmatrix} − 5 − 2 − 1 − 7

Welcome back. I am so glad you're here. We are given this matrix. I'm going to copy it down and reading down the first column negative eight, negative nine. And the second column negative six. Negative four. And we are to evaluate the second order determinant. Will we recall from previous lessons on second order determinants that we're going to take the term in the upper left hand corner and multiply it by the term in the bottom right hand corner. So negative eight times negative four which would be a positive 32. And then we take the term in the bottom left hand corner and multiplied by the term in the upper right hand corner. So negative nine times negative six which is a positive 54. And then we are going to subtract those terms from each other. So starting with 32 the first one and then subtracting the 2nd 1, 54 32 minus 54 is negative 22 that is our determinant. We look at our answer choices and that matches with answer choice. D. 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

1:12 minutes

Problem 7

Textbook Question

Evaluate each determinant in Exercises 1–10.
$\begin{vmatrix}-5 & -1 \\-2 & -7\end{vmatrix}$

Verified step by step guidance

Identify the matrix given as a 2x2 matrix: $\begin{bmatrix} -5 & -1 \\ -2 & -7 \end{bmatrix}$.

Recall the formula for the determinant of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, which is $ad - bc$.

Assign the values from the matrix to the variables: $a = -5$, $b = -1$, $c = -2$, and $d = -7$.

Substitute these values into the determinant formula: $(-5)(-7) - (-1)(-2)$.

Simplify the expression step-by-step to find the determinant value.

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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 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 Problem Solving

Evaluating determinants is a fundamental skill in algebra used to solve systems of equations, find inverses, and analyze linear transformations. Mastery of this concept aids in broader mathematical problem solving.

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