Problem 28 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$(A+E) D$$

Step-by-Step Solution

Answer

The operation is undefined; A and E have incompatible dimensions for addition.

Step 1: Check Dimensions of A and E

Matrix \(A\) has dimensions \(2 \times 3\) and matrix \(E\) has dimensions \(3 \times 3\).

Step 2: Determine if Addition is Possible

For matrix addition, both matrices must have the same dimensions. Since \(A\) is \(2 \times 3\) and \(E\) is \(3 \times 3\), they have different dimensions.

Step 3: State the Conclusion

The operation \((A + E)D\) cannot be performed because \(A + E\) is undefined due to incompatible dimensions.

Key Concepts

Matrix Addition

Matrix addition is a basic operation where two matrices of the same dimensions are summed element-wise. The dimensions of matrices play a crucial role in determining whether the addition can be performed.
To add two matrices, say matrix A and matrix B, each element of A must be added to the corresponding element of B. This means that both matrices must have an equal number of rows and columns.

\( A = \left[ \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array} \right] \) and \( B = \left[ \begin{array}{cc} b_{11} & b_{12} \ b_{21} & b_{22} \end{array} \right] \)
The result of adding A and B is \( A + B = \left[ \begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{array} \right] \)

Remember, if the matrices do not share the same dimensions, matrix addition cannot occur. As seen in the original exercise, matrices A (2 x 3) and E (3 x 3) have different dimensions and thus cannot be added.

Matrix Dimensions

Understanding matrix dimensions is pivotal when performing matrix operations. The dimension of a matrix is described by the number of its rows and columns, represented as \( m \times n \).

A matrix with 2 rows and 3 columns is stated as "two by three" or \( 2 \times 3 \).
An ease to remember: Rows go first, followed by columns when writing dimensions.

Checking dimensions helps in determining the possibility of operations such as addition or multiplication. The original example showed that since Matrix A is \( 2 \times 3 \) and Matrix E is \( 3 \times 3 \), they cannot be directly added.
In other matrices such as G which is \( 2 \times 2 \), matrix operations like multiplication require matching inner dimensions for the process to be possible.

Matrix Multiplication

Matrix multiplication is more complex compared to addition. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix.
Consider matrices X and Y, where X is an \( a \times b \) matrix and Y is a \( b \times c \) matrix. The resulting product is an \( a \times c \) matrix.

Each element of the product matrix is derived through "dot products" of corresponding row vectors from the first matrix and column vectors from the second matrix.
The dot product involves multiplying corresponding elements and summing them up.

For instance, the product of matrices \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \) and \( \left[\begin{array}{c} x \ y \end{array}\right] \) would be \( \left[\begin{array}{c} ax + by \ cx + dy \end{array}\right] \).
Matrix multiplication contrasts with addition due to its dependency on dimension alignment, allowing for a wider range of possible outcomes.