Problem 22 - 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. $$C-5 A$$

Step-by-Step Solution

Answer

The operation cannot be performed because \(C\) and \(A\) have different dimensions.

Step 1: Understand matrix dimensions

Before performing any operation, we should first understand the dimensions of each matrix involved. Matrix \(C\) is a \(2 \times 3\) matrix, and matrix \(A\) is a \(2 \times 2\) matrix.

Step 2: Check matrix compatibility

For matrix subtraction \(C - 5A\), both matrices must have the same dimensions. \(C\) is a \(2 \times 3\) matrix while \(A\) is \(2 \times 2\). Therefore, they do not have the same dimensions, and we cannot perform direct subtraction.

Key Concepts

Matrix Subtraction

Matrix subtraction is much like subtracting numbers, but performed on entries of matrices. Each element in the first matrix is subtracted by the corresponding element in the second matrix. Unlike simple numbers, matrices have strict rules for subtraction. Both matrices must have the same dimensions, meaning the same number of rows and columns. For example, if one matrix is size \(2 \times 3\), the other must also be \(2 \times 3\) for the subtraction to be valid. This requirement ensures each element of one matrix has a corresponding element in the other matrix to subtract from. If the dimensions don't match, like in our exercise with matrix \(C\) being \(2 \times 3\) and matrix \(A\) as \(2 \times 2\), subtraction cannot occur. To correct this, the dimensions must match, or you adjust by redefining matrices. Understanding compatibility through dimensional analysis is critical before performing any operations.

Matrix Dimensions

Matrix dimensions tell us the size of a matrix, specified by the number of rows and columns it has. This is written as \(m \times n\), where \(m\) is the number of rows, and \(n\) is the number of columns. For example, a matrix with 2 rows and 3 columns is a \(2 \times 3\) matrix. These dimensions dictate which operations can be performed on matrices. Matching dimensions are essential for operations like addition or subtraction, while multiplication needs a different kind of compatibility. Taking the example from our exercise, matrix \(C\) is \(2 \times 3\), making it incompatible with matrix \(A\), which is \(2 \times 2\), for subtraction. Understanding dimensions is the first step towards any matrix operation, ensuring each mathematical maneuver follows the rules of matrix algebra. Familiarity with matrix dimensions helps in recognizing valid operations quickly, avoiding mistakes in computational processes.

Matrix Compatibility

Matrix compatibility refers to whether matrices can engage in a particular operation based on their dimensions. Compatibility rules differ for addition, subtraction, and multiplication.

• Addition/Subtraction: Matrices must have the same dimensions, like two \(3 \times 3\) matrices.
• Multiplication: The number of columns in the first matrix must equal the number of rows in the second. For instance, a \(2 \times 3\) matrix can multiply with a \(3 \times 2\) matrix.

In operations with incompatible matrices, such as subtracting matrices of different sizes (like \(2 \times 3\) from \(2 \times 2\)), it's impossible to proceed without altering one of the matrices or adjusting the operation plan. Checking for compatibility involves assessing the dimensions to see if they align with the rules for the specific operation desired. Understanding these compatibility rules allows mathematicians and students to swiftly determine which matrix operations are feasible, streamlining problem-solving and ensuring accurate mathematical reasoning.