Products of Matrices The matrices A, B, and C are defined as...

Products of Matrices The matrices $A, B,$ and $C$ are defined as follows. $$ A=\left[ \begin{array}{rrrr}{1} & {0} & {6} & {-1} \\ {2} & {\frac{1}{2}} & {4} & {0}\end{array}\right] $$ $$ B=\left[ \begin{array}{llll}{1} & {7} & {-9} & {2}\end{array}\right] \quad C=\left[ \begin{array}{r}{1} \\ {0} \\ {-1} \\ {-2}\end{array}\right] $$ Determine which of the following products are defined, and calculate the ones that are. $$ \begin{array}{lll}{A B C} & {A C B} & {B A C} \\ {B C A} & {C A B} & {C B A}\end{array} $$

Products of Matrices The matrices $A, B,$ and $C$ are defined
as follows.
$$
A=\left[ \begin{array}{rrrr}{1} & {0} & {6} & {-1} \\ {2} & {\frac{1}{2}} & {4} & {0}\end{array}\right]
$$
$$
B=\left[ \begin{array}{llll}{1} & {7} & {-9} & {2}\end{array}\right] \quad C=\left[ \begin{array}{r}{1} \\ {0} \\ {-1} \\ {-2}\end{array}\right]
$$
Determine which of the following products are defined, and
calculate the ones that are.
$$
\begin{array}{lll}{A B C} & {A C B} & {B A C} \\ {B C A} & {C A B} & {C B A}\end{array}
$$

Key Concepts

Dot Product Calculation in Matrix Multiplication

Each entry in the resulting matrix from a multiplication is computed using the dot product of corresponding rows of the first matrix and columns of the second matrix. This involves multiplying corresponding elements and then summing the products. Mastery of the dot product is essential to carry out the actual calculations once the compatibility and order of multiplication are confirmed.

Order of Matrix Multiplication

The order in which matrices are multiplied greatly affects whether the multiplication is defined and the resulting matrix. The non-commutative property of matrix multiplication means that changing the order of the matrices may lead to undefined products or differing results. It is important to correctly arrange matrices such that the conditions for multiplication are satisfied.

Associative Property of Matrix Multiplication

The associative property states that when multiplying several matrices, the grouping of the matrices does not affect the final product, provided that the multiplication is defined. This is useful when evaluating products of three or more matrices because it allows flexibility in the order of operations during computation.

Matrix Multiplication Compatibility

Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix. This concept is crucial for determining whether a specific product of matrices can be computed. It requires checking the dimensions of each matrix in the sequence of multiplication to ensure that the operations can be performed.

Dimensions of Matrices

Understanding the dimensions (number of rows and columns) of matrices is fundamental in linear algebra. The dimensions dictate not only whether multiplication is defined but also what the dimensions of the resulting product will be. Keeping track of matrix sizes helps in planning the order of operations and verifying the validity of the multiplication sequence.

Transcript

00:01 In this problem, we're given three matrices, a, b, and c.

00:04 And we're going to want to start by determining which of our matrix products are going to be possible to calculate.

00:10 In order to do so, we're essentially going to need to look at the dimensions of each of our matrices.

00:16 So matrix a, we can see as two rows and four columns, one, two, three, four.

00:20 So it's a two by four matrix.

00:23 Matrix b has only one row and also four columns.

00:26 So it's a one by four matrix.

00:27 And finally, matrix c has four rows and only one column, so it's a four by one matrix.

00:35 So in order for two matrices to be able to be multiplied, we know that the first matrix must have dimensions m by n, and our second matrix must have dimensions n by l.

00:47 So the number of columns of our first matrix must be equal to the number of rows of our second matrix.

00:53 That's these two ends right here.

00:55 So we can go ahead and start with abc.

00:57 We know that a is a 2 by 4 matrix, and b is a 1 by 4 matrix.

01:03 So we can see that these two inner numbers are not equal.

01:07 So that tells us we're not going to be able to calculate the product a, b.

01:11 Next for a matrix acb, a is a 2 by 4 matrix, and c is a 4 by 1 matrix.

01:19 So we can see that these are going to be able to be multiplied, and we're going to be left with a 2 by 1 matrix.

01:25 So now we're multiplying a 2 by 1 matrix by matrix b, which is 1.

01:29 1 by 4.

01:30 So again, these two inner numbers are going to be equal.

01:33 So we are going to be able to compute acb, and it's going to give us a 2 by 4 matrix at the end.

01:40 Next, b -a -c.

01:42 Matrix b is a 1 by 4 matrix, and a is a 2 -4 matrix.

01:47 So we can see that we have a 4 and a 2, so they are not going to be able to be multiplied there.

01:54 For a b -c -a, b is a 1 -4 matrix, and c is a 4 -1 matrix.

01:59 So we will be able to multiply b and c, and it's going to result in a 1 by 1 matrix, that we're going to multiply by matrix a, which is 2 by 4.

02:08 So now the matrix b, c, is not going to be able to be multiplied by the matrix a, so we're not going to be able to calculate bca.

02:16 Next for matrix, c -a -b, c is a 4 -1 matrix, and a is a 2 -4 matrix.

02:23 So again, these inner numbers will not line up, and we can't calculate c -a -b.

02:27 And lastly, cba, c is a 4 by 1 matrix, and b is a 1 by 4 matrix...

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