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: $$ \begin{array}{ccccc} A & B & C & D & E \\ (4 \times 5) & (4 \times 5) & (5 \times 2) & (4 \times 2) & (5 \times 4) \end{array} $$ In each part, determine whether the given matrix expression is defined. For those that are defined, give the size of the resulting matrix. (a) $B A$ (b) $A C+D$ (c) $A E+B$ (d) $A B+B$ (e) $E(A+B)$ (f) $E(A C)$ (g) $E^{T} A$ (h) $\left(A^{T}+E\right) D$ Answer: (a) Undefined (b) $4 \times 2$ (c) Undefined (d) Undefined (e) $5 \times 5$ (f) $5 \times 2$ (g) Undefined (h) $5 \times 2$

Suppose that $A, B, C, D,$ and $E$ are matrices with the following sizes:
$$
\begin{array}{ccccc}
A & B & C & D & E \\
(4 \times 5) & (4 \times 5) & (5 \times 2) & (4 \times 2) & (5 \times 4)
\end{array}
$$
In each part, determine whether the given matrix expression is defined. For those that are defined, give the size of the resulting matrix.
(a) $B A$
(b) $A C+D$
(c) $A E+B$
(d) $A B+B$
(e) $E(A+B)$
(f) $E(A C)$
(g) $E^{T} A$
(h) $\left(A^{T}+E\right) D$
Answer:
(a) Undefined
(b) $4 \times 2$
(c) Undefined
(d) Undefined
(e) $5 \times 5$
(f) $5 \times 2$
(g) Undefined
(h) $5 \times 2$

Key Concepts

Matrix Transpose Operations

Transposing a matrix involves swapping its rows and columns, so that an m×n matrix becomes an n×m matrix. This operation is key when dealing with expressions that include transposed matrices because it changes the dimensions and, therefore, possibly the rules of compatibility for subsequent matrix operations.

Determining Matrix Product Dimensions

When multiplying matrices sequentially in an expression, one must carefully track the dimensions of the intermediate results. This involves applying the rule of multiplication compatibility repeatedly and ensuring that each multiplication step results in a matrix whose dimensions allow for the next operation. This concept helps in both verifying whether complex expressions are defined and calculating the size of the resulting matrix.

Matrix Multiplication Compatibility

For the product of two matrices to be defined, the number of columns in the first matrix must equal the number of rows in the second. This concept is crucial in determining whether a given multiplication can be performed and, if it can, it also dictates the dimensions of the resulting matrix – specifically, if matrix A is of size m×n and matrix B is of size n×p, then their product AB is defined and will have dimensions m×p.

Matrix Addition Compatibility

Matrix addition requires that the matrices involved have exactly the same dimensions. This ensures that corresponding entries can be added together. If the matrices do not share identical dimensions, the addition is undefined, which is an important criterion when evaluating expressions involving both addition and multiplication.

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