Problem 25 - Precalculus: Mathematical for Calculus

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as $$\begin{array}{l} A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right] \\ D=\left[\begin{array}{rrr} 7 & 3 \end{array}\right] & E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \\ F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$A D$$

Step-by-Step Solution

Answer

The multiplication of matrices \(A\) and \(D\) cannot be performed due to incompatible dimensions.

Step 1: Identify Matrix Dimensions

Matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(D\) is a \(1 \times 3\) matrix. For matrix multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix.

Step 2: Check Matrix Compatibility

Matrix \(A\) has 2 columns, while matrix \(D\) has 1 row. Since the columns of \(A\) do not match the rows of \(D\), the multiplication \(AD\) cannot be performed.

Step 3: Conclusion: Operation Not Possible

Since the dimensions of matrices \(A\) and \(D\) are incompatible for multiplication, the operation \(AD\) cannot be carried out.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is fundamental when dealing with matrix multiplication. A matrix's dimensions are given in terms of its rows (horizontal lines of elements) and columns (vertical lines of elements). This is often referred to as the "size" of the matrix. For example, matrix \(A\) given in the exercise is a \(2 \times 2\) matrix, which means it has 2 rows and 2 columns. On the other hand, matrix \(D\) is a \(1 \times 3\) matrix, having 1 row and 3 columns.

When approaching problems involving matrices, always check and note down the dimensions first:

• Identify the number of rows a matrix has.
• Determine the number of columns it has.

Matrix dimensions are crucial because they determine whether certain algebraic operations, such as addition or multiplication, can be performed. Easy memorization tip: a matrix described by "\(m \times n\)" has "rows \(m\)" and "columns \(n\).”

Matrix Compatibility

Matrix compatibility is the key to successfully multiplying two matrices together. To multiply any two matrices \(X\) and \(Y\), the number of columns in the first matrix \(X\) must equal the number of rows in the second matrix \(Y\). If this requirement is not met, the matrices are said to be incompatible for multiplication.

In the given exercise, we explored matrices \(A\) which is \(2 \times 2\) and \(D\) which is \(1 \times 3\). Here, matrix \(A\) has 2 columns, but matrix \(D\) has only 1 row. This mismatch (A’s 2 columns versus D’s 1 row) makes multiplication impossible for these specific matrices.

To summarize:

• Before attempting multiplication, always verify that the number of columns in the first matrix matches the number of rows in the second matrix.
• If they don't match, the matrices are incompatible for multiplication.

Ensuring compatibility prevents errors and helps in understanding matrix operations better.

Algebraic Operations

Algebraic operations involving matrices include addition, subtraction, and multiplication. Each of these operations follows specific rules and conditions. For multiplication, it was already discussed that dimensions must align. In the case of attempted multiplication of matrices \(A\) and \(D\), it was noted that they are incompatible, and so another algebraic operation was considered.

Here's a brief look at how these operations generally work:

• Addition/Subtraction: Matrices can only be added or subtracted if they are of the same dimensions. This means they must have identical numbers of rows and columns.
• Multiplication: Beyond checking dimensions for compatibility, the actual operation involves dot products of rows and columns to create a new matrix.

Understanding these rules allows for effective manipulation and utilization of matrices, paving the way for successful handling of complex mathematical problems.