Problem 10 - Precalculus: Mathematics for Calculus

Question

Perform the matrix operation, or if it is impossible, explain why. $$\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 1 & 0 \end{array}\right]-\left[\begin{array}{lll} 2 & 1 & -1 \\ 1 & 3 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is \( \left[ \begin{array}{ccc} -2 & 0 & 2 \\ 0 & -2 & 2 \end{array} \right] \).

Step 1: Understand the Matrix Dimensions

First, determine the dimensions of both matrices involved in the operation. The first matrix is a 2x3 matrix and the second matrix is also a 2x3 matrix.

Step 2: Determine if Subtraction is Possible

Matrix subtraction requires that both matrices have the same dimensions. Since both matrices are 2x3, we can proceed with the subtraction.

Step 3: Subtract the Matrices Element by Element

Subtract corresponding elements from the two matrices. The result of subtracting the given matrices is:\[\begin{array}{ccc}0-2 & 1-1 & 1-(-1) \1-1 & 1-3 & 0-(-2) \end{array}\]This simplifies to:\[\begin{array}{ccc}-2 & 0 & 2 \0 & -2 & 2\end{array}\]

Step 4: Write Down the Resulting Matrix

The matrix after subtraction is:\[\begin{array}{ccc}-2 & 0 & 2 \0 & -2 & 2\end{array}\]

Key Concepts

Matrix Operations

Matrix operations are fundamental techniques used in linear algebra that involve arithmetic calculations with matrices, such as addition, subtraction, and multiplication. These operations follow specific rules which ensure consistency and reliability in computations. Matrix subtraction is one such operation, and it involves the subtraction of corresponding elements from two matrices.

When performing matrix subtraction, it's critical to ensure that both matrices have the same dimensions. This prerequisite stems from the method of subtracting each element in one matrix directly from the corresponding element in the other matrix.

• Step 1: Check the dimensions of both matrices.
• Step 2: Subtract each element of the first matrix by the corresponding element of the second matrix.

These steps ensure that the operation yields a valid result, maintaining the integrity of mathematical principles and producing a new matrix with results displayed in the corresponding positions.

Matrix Dimensions

Matrix dimensions denote the size of a matrix, specifically the number of rows and columns it contains. This is usually expressed in the form of rows by columns, such as 2x3. Understanding matrix dimensions is vital because it helps determine the compatibility of matrices for various operations.

For example, when subtracting matrices, both matrices must have identical dimensions. This alignment allows for direct comparison and operation between each element's position in both matrices. The uniformity of structure is crucial to accurately perform element-wise calculations, like subtraction.

• If Matrix A is a 2x3 matrix, it has 2 rows and 3 columns.
• Matrix B must also be a 2x3 matrix to correctly perform subtraction.

Ensuring the matrices are compatible based on their dimensions prevents errors and allows for smooth mathematical operations, such as subtraction, addition, and beyond.

Element-wise Subtraction

Element-wise subtraction refers to the process of subtracting elements that occupy the same position within their respective matrices. It requires that both matrices share the same dimensions so that every element from one matrix can have a direct counterpart in the other matrix.

To perform element-wise subtraction, consider each element’s position within the individual matrices. The element from the first matrix is subtracted from the corresponding element in the second matrix. This results in a new matrix where each position contains the calculated result of the subtraction.

• Align each element of Matrix A with the corresponding element of Matrix B.
• Subtract the element in Matrix B from the element in Matrix A.
• Place the result in the new matrix at the same position.

The outcome is a new matrix, maintaining the same dimensions as the original matrices, making it easy to spot and verify each element's calculation. This process underscores the significance of matrix dimensions and compatibility in performing accurate matrix operations.