Suppose that A, B, C, D, and E are the following matrices: A=[ 3 0 -1

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

Suppose that $A, B, C, D$, and $E$ are the following matrices: $$ \begin{array}{c} A=\left[\begin{array}{cc} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{array}\right], B=\left[\begin{array}{cc} 4 & -1 \\ 0 & 2 \end{array}\right], C=\left[\begin{array}{ccc} 1 & 4 & 2 \\ 3 & 1 & 5 \end{array}\right] \\ D=\left[\begin{array}{ccc} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array}\right], \text { and } E=\left[\begin{array}{ccc} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array}\right] \end{array} $$ Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix. (a) $D+E$ (b) $D-E(c) 5 A(d)-7 C$ (e) $2 B-C$ $(f) 2 E-2 D(g)-3(D+2 E)$ $(h) A-A$ (i) $A B$ (j) $B A$ $(k)(3 E) D$ $(l)(A B) C(m) A(B C)(n)(4 B) C+2 B$ (o) $D-3 E$ $(p) C A+2 E$ $(q) 4 E-D(r) D D$

Suppose that $A, B, C, D$, and $E$ are the following matrices:
$$
\begin{array}{c}
A=\left[\begin{array}{cc}
3 & 0 \\
-1 & 2 \\
1 & 1
\end{array}\right], B=\left[\begin{array}{cc}
4 & -1 \\
0 & 2
\end{array}\right], C=\left[\begin{array}{ccc}
1 & 4 & 2 \\
3 & 1 & 5
\end{array}\right] \\
D=\left[\begin{array}{ccc}
1 & 5 & 2 \\
-1 & 0 & 1 \\
3 & 2 & 4
\end{array}\right], \text { and } E=\left[\begin{array}{ccc}
6 & 1 & 3 \\
-1 & 1 & 2 \\
4 & 1 & 3
\end{array}\right]
\end{array}
$$
Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix.
(a) $D+E$
(b) $D-E(c) 5 A(d)-7 C$
(e) $2 B-C$
$(f) 2 E-2 D(g)-3(D+2 E)$
$(h) A-A$
(i) $A B$
(j) $B A$
$(k)(3 E) D$
$(l)(A B) C(m) A(B C)(n)(4 B) C+2 B$
(o) $D-3 E$
$(p) C A+2 E$
$(q) 4 E-D(r) D D$

Key Concepts

Matrix Conformability

This concept entails that for two matrices to be added or subtracted, they must have the same dimensions, and for two matrices to be multiplied, the number of columns in the first must equal the number of rows in the second. This condition ensures that the operations are well defined and lays the groundwork for performing arithmetic operations on matrices.

Matrix Addition and Subtraction

Matrix addition and subtraction are performed by combining corresponding entries of matrices of the same dimensions. Each element in the resulting matrix is the sum or difference of the respective elements from the original matrices, making these operations straightforward and based on entry?wise computations.

Scalar Multiplication

Scalar multiplication involves multiplying every entry of a matrix by a scalar (a real or complex number). This operation distributes over the matrix and is used to scale the matrix, preserving its structure while modifying its entries proportionally.

Matrix Multiplication

Matrix multiplication is defined as the dot product of the rows of the first matrix with the columns of the second matrix. The product of an m×n matrix and an n×p matrix is an m×p matrix, and this operation is not generally commutative, meaning that the order of multiplication matters.

Order of Operations in Matrix Arithmetic

When multiple matrix operations are involved, it is essential to follow the proper order of operations. Parentheses indicate which operation should be performed first, and operations like scalar multiplication must be performed prior to or concurrently with addition or subtraction as dictated by the expression, to ensure accurate computation.

Associative and Distributive Properties

Matrix operations follow certain algebraic properties such as associativity (for matrix multiplication and addition) and distributivity (scalar multiplication distributing over matrix addition). These properties facilitate the simplification and computation of more complex matrix expressions while ensuring consistency in the results.

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