Use the determinant theorems to evaluate each determinant. ∣ 1 6 7 0 6 7 0 0 9 ∣ \left| \begin{matrix} 1 & 6 & 7 \\ 0 & 6 & 7 \\ 0 & 0 & 9 \end{matrix} \right|

Hello, today we're gonna be finding the determinant of the three by three matrix. Now the three by three matrix given to us is 11 06 00 14. Now there's one thing to note about the current matrix, the current matrix is in the form of an upper diagonal matrix. And the reason for this is because the lower corner of this matrix contains elements of zero. So only the upper right form of the matrix contains nonzero elements which kind of forms this upper trian triangle inside the inner matrix. So we're going to classify this matrix as an upper triangular matrix. And since this is an upper triangular matrix, that means the determinant is going to be the product of the diagonal of the matrix. So in order to get the determinant of this three by three matrix, we're gonna be multiplying five with six with and five times six times 14 is going to give us the value of 420. And this is going to be the determinant of this three by three matrix. And with that being said, the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

1:38 minutes

Problem 53

Textbook Question

Use the determinant theorems to evaluate each determinant.
$∣ \begin{matrix} 1 & 6 & 7 \\ 0 & 6 & 7 \\ 0 & 0 & 9 \end{matrix} ∣$

Verified step by step guidance

Identify the size and structure of the given determinant matrix to understand which determinant theorems can be applied effectively.

Recall key determinant theorems such as: the determinant of a matrix with two identical rows is zero, swapping two rows changes the sign of the determinant, and the determinant of a triangular matrix is the product of its diagonal entries.

Apply row operations that simplify the matrix to a form where the determinant is easier to calculate, keeping track of how each operation affects the determinant value according to the theorems.

Use the properties that multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant, to further simplify the matrix if needed.

Once the matrix is in a simplified form (such as upper triangular), calculate the determinant by multiplying the diagonal entries, and adjust the sign or scale of the determinant based on the row operations performed.

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Key Concepts

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

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties such as invertibility. It can be calculated using various methods, including expansion by minors or row operations, and is essential for solving systems of equations and understanding matrix behavior.

Determinant Theorems

Determinant theorems are rules that simplify the calculation of determinants, such as the effect of row swaps, scalar multiplication of rows, and adding multiples of one row to another. These theorems help reduce complex matrices to simpler forms without changing or by predictably changing the determinant.

Row Operations and Their Impact on Determinants

Certain row operations affect the determinant in specific ways: swapping two rows multiplies the determinant by -1, multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant. Understanding these effects is crucial for efficient determinant evaluation.

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