Problem 89 - Precalculus

Question

Explain the mistake that is made. $$\text { Multiply }\left[\begin{array}{ll} 3 & 2 \\\1 & 4\end{array}\right]\left[\begin{array}{ll}-1 & 3 \\\\-2 & 5\end{array}\right]$$ Solution: Multiply corresponding elements. Simplify. $$\left[\begin{array}{ll}3 & 2 \\\1 & 4 \end{array}\right]\left[\begin{array}{ll}-1 & 3 \\\\-2 & 5\end{array}\right]=\left[\begin{array}{ll} (3)(-1) & (2)(3) \\\\(1)(-2) & (4)(5)\end{array}\right]$$This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was incorrectly treating matrix multiplication as element-wise multiplication instead of using the dot product approach.

Step 1: Understand Matrix Multiplication

Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second matrix. For a valid multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

Step 2: Setup Matrix Dot Products

For two matrices \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \), the result is \( AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} \).

Step 3: Apply Dot Product Method

Apply this method to the given matrices: \( A = \begin{bmatrix} 3 & 2 \ 1 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & 3 \ -2 & 5 \end{bmatrix} \).Calculate each element:- First row, first column: \( (3)(-1) + (2)(-2) = -3 - 4 = -7 \).- First row, second column: \( (3)(3) + (2)(5) = 9 + 10 = 19 \).- Second row, first column: \( (1)(-1) + (4)(-2) = -1 -8 = -9 \).- Second row, second column: \( (1)(3) + (4)(5) = 3 + 20 = 23 \).

Step 4: Write the Correct Result

The correct product of the two matrices is: \( \begin{bmatrix} -7 & 19 \ -9 & 23 \end{bmatrix} \).

Key Concepts

Dot Product

The dot product is a key operation in matrix multiplication. To calculate it, you take two sequences of numbers (vectors), multiply corresponding elements, and sum the results.

Matrix multiplication relies heavily on this operation. In the context of matrices, consider taking one row from the first matrix and one column from the second.

• Multiply each element from the row vector by the corresponding element in the column vector.
• Add up the results to get a single number.

This resulting number fills a spot in the new matrix being created from multiplication.

Consider a simple example: if you are computing the dot product for the first row of one matrix and the first column of another, these mapped pairs produce a value for the first row and column position of the resulting matrix. Mastering the dot product step is crucial for any matrix operation.

Matrix Dimensions

Understanding matrix dimensions is essential before performing any matrix operations. Matrix dimensions are written as 'number of rows × number of columns'.

For matrix multiplication to work, the number of columns in the first matrix must equal the number of rows in the second matrix. This means only certain matrices can be multiplied together.

• For example, if matrix A is a 2x3 matrix and matrix B is a 3x2 matrix, multiplication is doable. Matrix A has 2 rows and 3 columns, while matrix B has 3 rows and 2 columns.
• The result of multiplying these matrices will be a new matrix with dimensions of 2x2.

Correctly identifying these dimensions safeguards against missteps and helps progress in solving larger matrix operations. Always double-check these dimensions for compatibility.

Matrix Operations

Matrix operations involve various mathematical processes you perform on matrices. Among these, matrix addition, subtraction, and multiplication are common.

In addition and subtraction, matrices must be of the same dimensions. Every element of one matrix is added to or subtracted from the corresponding element of the other matrix. This differs from multiplication in both its method and requirement.

• For multiplication, as discussed, the inner dimensions must match because you're computing dot products.
• The resulting matrix has the number of rows of the first matrix and the number of columns of the second.

Proper understanding of each type of matrix operation helps clarify what is possible mathematically and facilitates accurate computation.