Suppose that A, B, and C are the following matrices and that a=4 and b=7. A=[ 1 5 2

step 1 Given these matrices A, B, and C, let's multiply A times B. So remember when we're multiplying m

Suppose that A, B, and C are the following matrices and that...

Key Concepts

Compatibility of Scalar Multiplication with Matrix Multiplication

Scalar multiplication interacts predictably with matrix multiplication, meaning that scaling the product of matrices by a scalar is equivalent to scaling any one of the matrices before performing the multiplication; for instance, a(BC) = (aB)C = B(aC). This makes it easier to factor and rearrange terms in expressions involving both operations.

Concept of Matrix Subtraction as Addition of the Negative Matrix

Matrix subtraction is defined as the addition of the additive inverse; that is, B - C is the same as B + (-C). This perspective reinforces the commutative property of addition, as reordering the terms (such as -C + B) does not affect the outcome, and provides a basis for understanding and applying algebraic manipulations with matrices.

Distributive Property of Matrix Multiplication over Matrix Addition

This property explains that matrix multiplication distributes over matrix addition from both the left and right sides, such as A(B + C) = AB + AC and (B + C)A = BA + CA. This rule is important for expanding and factoring expressions that involve the multiplication of matrices and the addition of matrices.

Associative Property of Matrix Multiplication

For matrices, the associative property of multiplication indicates that for any three matrices where the product is defined, (AB)C = A(BC). This property is crucial for simplifying expressions involving multiple matrix multiplications and is distinct from the non-commutative nature of matrix multiplication.

Distributive Property of Scalar Multiplication over Matrix Addition/Subtraction

This concept shows that a scalar multiplied by the sum or difference of matrices can be distributed; that is, a(B + C) = aB + aC and a(B - C) = aB - aC. It highlights the compatibility between scalar multiplication and matrix addition, allowing for easier manipulation and simplification of expressions.

Associative Property of Matrix Addition

This property states that when adding three matrices, the grouping does not affect the result; that is, (A + B) + C = A + (B + C). It is a fundamental aspect of matrix addition that ensures the sum remains the same regardless of how the matrices are paired during addition.

Transcript

00:01 Given these matrices a, b, and c, let's multiply a times b.

00:07 So remember when we're multiplying matrices, first we need to check that the multiplication is defined.

00:15 So what we check is the dimensions of both of these matrices.

00:21 So if the number of columns in the first matrix is equal to the number of rows in the second matrix, this multiplication is defined.

00:30 And in this case it is.

00:32 So what we're going to do is multiply the row of the first matrix to the column of the second and add those.

00:39 So we would have 1 times 5, which is 5, plus 2 times negative 6, negative 12.

00:50 And then in this first row second column, we would have 2 times 0, 0 plus 2 times 7, which is 14.

01:03 Now in this second row first column, 3 times 5, 15, plus negative 4 times negative 6, which is 24.

01:16 And then second row, second column, 3 times 0 is 0, plus negative 4 times positive 7 is negative 28.

01:29 And when we add those together, we end up with negative 7, 14, 39, negative 28.

01:46 Now let's multiply that product ab times matrix c.

01:51 Matrix a, b was a 2 by 2, and matrix c is a 2 by 3.

02:03 So since the number of columns in the first matrix equals the number of rows in the second, this multiplication is defined.

02:13 So multiplying this first row and first column for this element, negative 7 times 1, negative 7, plus 14 times 2, which goes this 28.

02:27 First row's second column, so we would have negative 7 times negative 3, which is 21, plus 14 times 6, which is 84.

02:42 And first row, third column, negative 7 times 4, negative 28, plus 14 times negative 5 gives us negative 70.

02:59 Second row, first column, so 39 times 1 is 39, plus negative 20.

03:08 28 times 2 is negative 56.

03:20 And let's see, second row, second column, 39 times negative 3, negative 117, plus negative 28 times 6, negative 168.

03:37 And then second row, third column, 39 times 4, is, 156 plus negative 28 times negative 5 is a positive 140.

03:58 So adding those up, we end up with 215, negative 98.

04:09 And in our second row, we would have negative 17, negative 285, negative 285, now let's take a look at the multiplication of matrix b times matrix c.

04:23 And again notice that there are two columns in matrix b and two rows in matrix c, so this multiplication is defined...

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