In Problems 7-14, find the value of each determinant.

step 1 Hello. So here we are taking the determinant of this 3 by 3 matrix. So we are going to expand al

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Key Concepts

Determinant

A determinant is a scalar value that can be computed from the elements of a square matrix. It encapsulates key properties of the matrix, such as whether the matrix is invertible, and it reflects how the matrix scales volumes in the space it acts upon. In linear algebra, the determinant plays a crucial role in solving systems of linear equations and understanding linear transformations.

3x3 Matrix Determinant Calculation

For a 3x3 matrix, the determinant is typically calculated using either the rule of Sarrus or the method of cofactor expansion. The rule of Sarrus involves summing the products of the diagonals of an augmented matrix and subtracting the products of the diagonals in the opposite direction. This method provides a straightforward procedure to obtain the determinant for a 3x3 matrix.

Cofactor Expansion

Cofactor expansion is a general method used to compute the determinant of a matrix by expanding it along a row or column. This method involves calculating the minors for each element of the chosen row or column and multiplying them by their corresponding cofactors (which include a sign factor dependent on the element's position). Cofactor expansion is particularly useful for matrices larger than 3x3, although it can also be applied to 3x3 matrices.

Transcript

0:00 Hello.

00:01 So here we are taking the determinant of this 3 by 3 matrix.

00:04 So we are going to expand along the first row.

00:07 So what we do is we look at the first entrance is 3 and then cross out the row and column, which we're in, and then take the determinant of the corresponding 2 by 2 matrix.

00:17 So we have 3 and then times the determinant of, well, we get 4, 0, negative 3, 1, and then it's minus the second entry times the term of its minor.

00:28 So it's but it's minus the second entry.

00:30 Here in this first row is negative 9.

00:32 So minus a negative 9, right? what we have is minus a negative 9.

00:36 Let me write that minus a negative 9.

00:38 That's going to be plus 9, right? and times the determinant of its minor.

00:42 So cross out the row and column.

00:43 And we are left with 1 .083, or 81.

00:51 Okay.

00:52 And then we have plus the last entry, which is 4, and then times the determinant of its 2 by 2.

00:59 So cross out the row again, row and column.

01:01 And look at 1, 4, 8, negative 3.

01:07 Okay, so what do we have now? what we have three times? well, the determinant of a 2 by 2 is just the product of the main diagonal minus the product of the other diagonal.

01:16 So you have three times, well, it's 4 times 1, which is 4, and then minus negative 3 times 0...

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