Scalar Multiplication
Assume generic matrix C
To multiply C by a scalar k, simply multiply each entry by k.
Here is an example with numerical entries and a specific scalar.
Scalar Multiplication Practice: Given matrix A, find each of the following.
#1) 2A
#2) -10A
Matrix Multiplication
How To Multiply Two Matrices
To multiple two matrices, you need to do the "dot product" of the rows and columns. Here's an example:
To find the entry in row 1 column 1, (2)(1) + (3)(4) + (4)(7) = 2 + 12 + 28 = 42
To find the entry in row 1 column 2,(2)(5) + (3)(8) + (4)(8) = 10 + 24 + 32 = 66
To find the entry in row 2 column 1, (6)(1) + (-1)(4) + (0)(7) = 6 + (-4) + 0 = 2
To find the entry in row 2 column 2, (6)(5) + (-1)(8) + (0)(8) = 30 + (-8) + 0 = 22
When It Is Possible To Multiply Two Matrices?
Two matrices A and B can be multiplied together, AB if the first matrix has the same number of of columns as the second matrix has rows. For example if A is size m x n and n x q is the size of B, then the matrices can be multiplied.
Matrices That Can Be Multiplied
When comparing A and B, the sizes side by side are 1 x 4 and 4 x 1. Since the middle two numbers are the same, the matrices can be multiplied. So AB can be found and its size is 1 x 1.
When comparing D and E, the sizes side by side are 3 x 3 and 3 x 2. Since the middle two numbers are the same, the matrices can be multiplied. So DE can be found and its size is 3 x 2.
When comparing F and G, the sizes side by side are 3 x 4 and 4 x 2. Since the middle two numbers are the same, the matrices can be multiplied. So FG can be found and its size is 3 x 2.
Matrices That Can NOT be Multiplied
When comparing K and L, the sizes side by side are 1 x 4 and 1 x 4. Since the middle two numbers are NOT the same, the matrices can NOT be multiplied.
When comparing M and N, the sizes side by side are 2 x 5 and 1 x 4. Since the middle two numbers are NOT the same, the matrices can NOT be multiplied.
Matrix Multiplication Practice: Given the matrices, answer each question.
#1) AB
#2) BA
#3) CD
#4) DC
Homework
Scalar Multiplication
Find each product
#2) 5B
#3) 100C
#4) -2D
#5) -B
Matrix Multiplication?
Determine which products can be found. Justify your answer. If the product can be found, state the dimension of the matrix product.#2) BA
#3) BC
#4) CB
#5) CD
#6) DC
Matrix Multiplication
Find each product given the matrices.
#2) AE
#3) BD
#4) CD
#5) DC
#6) DF
#7) EB
#8) FE
Answers Scalar Multiplication
Answers Matrix Multiplication?
#1) Yes because A is 3x2 and B is 2x4. Since the 2's are the same, they can be multiplied with the resultant matrix being 3x4.#2) No because B is 2x4 and A is 3x2. Since the 4 and 3 aren't the same, they can NOT be multiplied.
#3) Yes because B is 2x4 and C is 4x1. Since the 4's are the same, they can be multiplied with the resultant matrix being 2x1.
#4) No because C is 4x1 and B is 2x4. Since the 1 and 2 aren't the same, they can NOT be multiplied.
#5) Yes because C is 4x1 and D is 1x4. Since the 1's are the same, they can be multiplied with the resultant matrix being 4x4
#6) Yes because D is 1x4 and C is 4x1. Since the 4's are the same, they can be multiplied with the resultant matrix being 1x1.
Answers Matrix Multiplication
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