Problem 4 - Algebra and Trigonometry

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 result of the matrix operation is \( \begin{array}{ccc} -2 & 0 & 2 \\ 0 & -2 & 2 \end{array} \).

Step 1: Check Matrix Dimensions

First, ensure that both matrices have the same dimensions. The first matrix is a 2x3 matrix, and the second matrix is also a 2x3 matrix. Therefore, the subtraction operation is possible.

Step 2: Subtract Corresponding Elements

Subtract each element of the first matrix from the corresponding element of the second matrix. Perform the operation element-wise: \[ \begin{array}{ccc} 0-2 & 1-1 & 1-(-1) \ 1-1 & 1-3 & 0-(-2) \end{array} \]

Step 3: Simplify the Subtraction

Calculate the result of each element-wise subtraction: \[ \begin{array}{ccc} -2 & 0 & 2 \ 0 & -2 & 2 \end{array} \]

Step 4: Present the Result

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

Key Concepts

Matrix Subtraction

Matrix subtraction involves taking two matrices of the same dimensions and subtracting their corresponding elements. This operation is only successful when both matrices have the same number of rows and columns. This means that each element in one matrix has a direct counterpart in the other matrix. If the matrices do not have the same dimensions, subtraction cannot be performed, as there would be no matching elements to subtract from one another.

To subtract two matrices, you simply subtract each element of the first matrix from the corresponding element of the second matrix. For example, given two matrices \(\left[\begin{array}{cc}3 & 8 \2 & 5\end{array}\right]\)and \(\left[\begin{array}{cc}1 & 6 \4 & 2\end{array}\right]\),the subtraction operation would produce:\(\left[\begin{array}{cc}3-1 & 8-6 \2-4 & 5-2\end{array}\right] = \left[\begin{array}{cc}2 & 2 \-2 & 3\end{array}\right]\).

Remember, if the matrices do not match in size, subtraction is not defined.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing operations like matrix subtraction. A matrix's dimensions are defined by its number of rows and columns. These dimensions are often expressed as \(m \times n\), where \(m\) represents rows and \(n\) represents columns.

For example, a matrix:\(\left[\begin{array}{ccc}1 & 2 & 3 \4 & 5 & 6\end{array}\right]\)is described as a 2x3 matrix, meaning it has 2 rows and 3 columns. When performing matrix subtraction, verifying that both matrices involved have identical dimensions is the first critical step. If they do not, the operation will be invalid because you won't have a one-to-one correspondence of elements between the two matrices to perform subtraction.

In the original exercise, both matrices were 2x3, confirming that matrix subtraction was indeed possible.

Element-wise Operations

Element-wise operations refer to applying mathematical operations individually to each corresponding element in a pair of matrices. This approach is employed in operations such as matrix addition, subtraction, and multiplication by a scalar.

In the context of matrix subtraction, each element of the first matrix is subtracted from its corresponding element in the second matrix. For instance, consider two matrices for subtraction:\(\left[\begin{array}{cc}7 & 4 \3 & 9\end{array}\right]\)and \(\left[\begin{array}{cc}5 & 2 \3 & 6\end{array}\right]\).The operation becomes \(\left[\begin{array}{cc}7-5 & 4-2 \3-3 & 9-6\end{array}\right] = \left[\begin{array}{cc}2 & 2 \0 & 3\end{array}\right]\).

Element-wise operations simplify matrix arithmetic by breaking it down into straightforward calculations for each pair of corresponding elements. This step-by-step process ensures computational accuracy and aids in understanding how individual elements contribute to the overall result.