Evaluate each determinant in Exercises 1–10. ∣ − 4 1 5 6 ∣ \begin{vmatrix}-4 & 1 \\5 & 6\end{vmatrix} − 4 5 1 6

Welcome back. I am so glad you're here. We are given this two by two. Matrix. I'm going to copy down the first column negative 82 and then the second column 74. And we are asked to evaluate the second order determinant. Will we recall from previous lessons on second order determinants. We're going to take the term in the upper left hand corner and multiplied by the term in the bottom right hand corner. So negative eight times four, which is negative 32. And then we will take the term in the bottom left hand corner and multiply it by the term in the upper right hand corner. So two times seven which is 14. And then we are going to subtract those two terms from each other. So the first term negative 32 minus the second term, 14, negative 32 minus 14 gives us a negative 46 and this is our determinant. We look at our answer choices and this matches with answer choice. A 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

1:10 minutes

Problem 3

Textbook Question

Evaluate each determinant in Exercises 1–10.
$\begin{vmatrix}\)-4 & 1 \\5 & 6\(\end{vmatrix}$

Verified step by step guidance

Identify the matrix given as $ \begin{pmatrix} -4 & 1 \\ 5 & 6 \end{pmatrix} $.

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

Substitute the values from the matrix into the formula: $ a = -4, b = 1, c = 5, d = 6 $.

Calculate the product of the diagonal elements: $ a \times d = (-4) \times 6 $.

Calculate the product of the off-diagonal elements: $ b \times c = 1 \times 5 $, then subtract this from the previous 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' are in the first row, 'c' and 'd' in the second. Correctly identifying these values is essential for accurate determinant calculation.

Applications of Determinants

Determinants are used to solve systems of linear equations, find matrix inverses, and analyze geometric transformations. Evaluating determinants is a foundational skill in linear algebra and college algebra.

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