Let A=[ 2 5 -3 1 ] and B=[ 4 -5 3 k ] . What value(s) of k, if any, will

step 1 Okay, for this problem, we're given two matrices, A and B. We want to find what values, if any,

Let A=[ 2 5 -3 1 ] and B=[ 4 -5 3 ...

Key Concepts

Solving Parameterized Systems of Equations

In linear algebra, it is common to encounter equations that include unknown parameters within matrices. Solving for these parameters involves formulating a system of equations based on the equality of corresponding matrix entries and then applying techniques such as substitution, elimination, or matrix methods to find the conditions under which the equations hold true. This systematic approach is crucial when determining the specific values that allow certain matrix relations, such as commutativity, to occur.

Equating Corresponding Entries of Matrices

When two matrices are equal, each entry in one matrix must be equal to the corresponding entry in the other matrix. This principle is used to set up equations that allow one to solve for unknown parameters. By comparing each corresponding pair of entries, a system of equations is formed, which can then be analyzed using standard algebraic techniques.

Commutativity in Matrix Multiplication

Unlike scalar multiplication, matrix multiplication generally does not satisfy the commutative property, meaning that for most matrices, the product of two matrices depends on the order in which they are multiplied. Determining when two matrices commute (i.e., when their product is the same regardless of the order) is a nuanced topic in linear algebra and often leads to conditions or constraints on the entries of the matrices.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that involves taking the dot product of rows from the first matrix with columns of the second matrix. This operation is defined by summing the products of corresponding elements, and the result is a new matrix whose entry in a given position represents the combined effect of the corresponding row and column. The procedure is essential for understanding how linear transformations combine in sequence.

Transcript

00:01 Okay, for this problem, we're given two matrices, a and b.

00:04 We want to find what values, if any, of k will make a times b the same as b times a.

00:12 So the way we want to do, the way that we want to do this is first calculate a times b.

00:18 So that's going to be doing the row column method.

00:23 We have 2 times 4 plus 3 times 5, 3 times 15, so we have 8 plus 15 in there.

00:28 Then for the next row we have negative 3 times 4, which is going to be negative 12, plus 3 times 1.

00:38 Then for the first row, second column element, we have 2 times negative 5, which is negative 10, plus 5k.

00:48 Then we have, for the last element, negative 3 times negative 5, which is going to be positive 15 plus k.

00:57 So that's our ab.

00:59 Actually, i guess i should calculate out exactly those values.

01:03 A plus 15 is 23, negative 12 plus 3 is 9, but we can't actually say the negative 10 plus 5k.

01:11 Oh, actually, i guess the factor, a common factor of 5.

01:15 That turns into 5 times k minus 2.

01:22 And then we have 15 plus k.

01:25 Nothing fancy we can do there.

01:28 Next, looking at doing b times a.

01:32 For the logic of it i'd like to have the matrices next to each other visibly makes it a little bit easier to keep track of things so we have four times two going to be eight plus negative five times negative three so we have eight plus 15 looks familiar then we have for the next row we're going to have yeah sorry we're going to have three times two so so 6 minus 3 times k.

02:13 Then for the first row, second column element, we'll have 4 times 5.

02:19 That's 20 minus 5...

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