Suppose that A, B, C, D, and E are matrices with the...

Suppose that $A, B, C, D,$ and $E$ are matrices with the following sizes: A) $(4 \times 5)$ $\quad$ B) $(4 \times 5)$ $\quad$ C) $(5 \times 2)$ $\quad$ D) $(4 \times 2)$ $\quad$ E) $(5 \times 4)$ In each part, determine whether the given matrix expression is defined. For those that are defined, give the size of the resulting matrix. (a) $C D^{T}$ (b) $D C$ (c) $B C-3 D$ (d) $D^{T}(B E)$ (e) $B^{T} D+E D$ (f) $B A^{T}+D$

Suppose that $A, B, C, D,$ and $E$ are matrices with the following sizes:
A) $(4 \times 5)$ $\quad$ B) $(4 \times 5)$ $\quad$ C) $(5 \times 2)$ $\quad$ D) $(4 \times 2)$ $\quad$ E) $(5 \times 4)$
In each part, determine whether the given matrix expression is defined. For those that are defined, give the size of the resulting matrix.
(a) $C D^{T}$
(b) $D C$
(c) $B C-3 D$
(d) $D^{T}(B E)$
(e) $B^{T} D+E D$
(f) $B A^{T}+D$

Key Concepts

Transpose Operation

Taking the transpose of a matrix involves swapping its rows and columns. If a matrix is of size m x n, its transpose is of size n x m. This operation often changes the multiplication compatibility with other matrices, making it an important concept when evaluating expressions involving transposed matrices.

Dimension Compatibility

Dimension compatibility refers to the necessity of matching appropriate dimensions for different matrix operations, such as multiplication (where inner dimensions must match) and addition (where both matrices must share the same size). This fundamental concept underpins the process of determining whether a given matrix expression is defined.

Matrix Multiplication

Matrix multiplication is defined when the number of columns of the first matrix equals the number of rows of the second matrix. If this condition is met, the resulting matrix will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix. This concept is crucial for determining the validity and dimensions of products in matrix expressions.

Matrix Addition

Matrix addition is only possible when both matrices have exactly the same dimensions. Each entry in the first matrix is added to the corresponding entry in the second matrix to form a new matrix of the same size. This concept ensures that terms being added are dimensionally compatible in matrix expressions.

Transcript

00:01 So we have matrices, a, b, c, d, and e here, which the sizes are, well, we have a 4 by 5 matrix, and then we have another 4 by 5 matrix.

00:14 Oops, 4 by 5.

00:22 Okay, and then we have a 5 by 2 and a 4 by 2 and a 5 by 4.

00:35 Okay, so here are the sizes of matrices a, b, c, d, and e, respectively.

00:43 So, part a, well, the size of matrix c is 5 by 2, and the size of matrix d is 4 by 2.

00:53 So the transpose of matrix d is just obtained by interchanging the rows and the columns of d.

01:01 So therefore, d transpose is going to be, well, it's going to be.

01:07 To be a 2x4 matrix.

01:11 Okay.

01:12 And then the product here, well, c times d transpose, to see if this is defined, well, we have to make sure that the number of columns in c is same as the number of rows in d transpose.

01:29 So here the number of columns in c is equal to the number of rows in d transpose.

01:34 So hence the product, c times d transpose, is defined, and the size of c times the transpose will be a 5x4 matrix.

01:49 Right.

01:50 And then we go on to part b.

01:52 So part b, we look at matrix c, which is a 5x2 matrix, and matrix d, which is 4 by 2.

02:03 So to find the product, d times c, while we multiply d, which is 5 by 2, and then multiply that by a 4 by 2.

02:19 So if we multiply 5 by 2, and then we multiply that by a 4 by 2 matrix, well, the number of columns in the first matrix is not the same as the number of rows in the second matrix.

02:33 The inside numbers here are not the same.

02:36 Therefore, this product, right, the product here would be d times c is not defined.

02:43 So is not defined, right? since the number of columns in c is not equal to the number of rows in d.

02:53 So not defined.

03:00 All right.

03:02 Now part c.

03:05 Well, in part c, we look at matrix b, which is a 4 by 5 matrix.

03:10 And matrix c, which is 5 by 2, and matrix d, which is 4 by 2.

03:15 So first to find the product, bc, well, for bc, we check again the number of columns in b, with the number of rows in c.

03:27 So here we are equal.

03:29 So here the product, bc, is possible.

03:32 It is defined, and the size of b .c would be a 4 by 2 matrix.

03:39 Okay, and then while matrix d is 4 by 2, and the multiplication of matrix d with a scalar, 3 does not affect the size of the matrix, so the size of bc is equal to the size of 3d.

03:57 Hence, bc minus 3d, right? we subtract bc minus 3d.

04:04 These are the same size, so yes, this is defined, and the size of bc minus 3d.

04:09 Minus 3d will be a 4 by 2.

04:17 All right.

04:19 Then going on to part d.

04:21 So in part d, we look at matrix b, which is a 4 by 5 matrix, and the matrix d, which is 4 by 2, and matrix e, which is 5 by 4.

04:32 So here first to find the product b, or we check the number of columns in b with the number of rows in e.

04:40 So here the number of columns in b is equal.

04:43 To the number of rows in e.

04:45 Therefore, b .e.

04:48 B .e is defined...