Aproperty of determinants is given (A and B are square matrices). State how the property has b

Aproperty of determinants is given (A and B are square...

Aproperty of determinants is given $(A \text { and } B$ are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If $B$ is obtained from $A$ by adding a multiple of a row of $A$ to another row of $A$ or by adding a multiple of a column of $A$ to another column of $A$, then $|B|=|A|$. $$\text { (a) }\left|\begin{array}{rr}1 & -3 \\5 & 2\end{array}\right|=\left|\begin{array}{rr} 1 & -3 \\0 & 17\end{array}\right|$$ (b) $\left|\begin{array}{rrr}5 & 4 & 2 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|=\left|\begin{array}{rrr}1 & 10 & -6 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|$

Aproperty of determinants is given $(A \text { and } B$ are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results.
If $B$ is obtained from $A$ by adding a multiple of a row of $A$ to another row of $A$ or by adding a multiple of a column of $A$ to another column of $A$, then $|B|=|A|$.
$$\text { (a) }\left|\begin{array}{rr}1 & -3 \\5 & 2\end{array}\right|=\left|\begin{array}{rr}
1 & -3 \\0 & 17\end{array}\right|$$
(b) $\left|\begin{array}{rrr}5 & 4 & 2 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|=\left|\begin{array}{rrr}1 & 10 & -6 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|$

Key Concepts

Determinant

A determinant is a scalar value computed from a square matrix that summarizes certain properties of the matrix, such as its invertibility, linear independence of its rows or columns, and the scaling factor of the linear transformation defined by the matrix. It is a fundamental concept in linear algebra and is calculated using specific rules that depend on the size of the matrix.

Elementary Row and Column Operations

One important property of determinants is that adding a multiple of one row (or column) to another does not change the value of the determinant. This invariance under such row or column operations simplifies the computation of determinants by allowing matrices to be transformed into easier-to-compute forms, such as triangular matrices, without affecting the determinant’s value.

Transcript

00:01 In this problem, we're working with a property for determinants of a square matrix.

00:06 We're given a square matrix, a, and if a square matrix b is obtained by adding a multiple of one row or column to another row or column in a, then a and b will have equal determinants.

00:23 So let's take a look to see if b was obtained in that manner.

00:27 You can see that the first two, first row is the same in both matrix b and matrix a, 1 negative 3.

00:36 But let's look at the second row, and you can see the second row in matrix b is different from the second row in matrix a.

00:44 So in order to obtain that row, it looks like in matrix a, row 1 was multiplied by 5, and then that was added to row 2.

00:56 So in this specific case, 5 times 1 is 5 plus 0 is 5.

01:04 5 times negative 3 is negative 15 plus 17 is 2.

01:08 So this is how row 2 in matrix b was obtained by multiplying row 1 in matrix a and adding it to row 2.

01:21 So now what we want to do is look at the determinants to see that they are equal.

01:26 And this one is pretty easy to do manually.

01:30 You could use a graphing calculator as well.

01:32 But let's go ahead and write down here the determinant of matrix b.

01:38 And the determinant of matrix b is going to be looking down the diagonal, 1 times 2 minus 5 times negative 3.

01:49 And that's going to give us 2 plus 15 is 17.

01:54 So now let's take a look at the determinant for matrix.

01:57 A, and in matrix a, we're going to multiply 1 times 17 minus 0 times negative 3, and that gives us 17...

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