Question

describe and correct the error a student made in multiplying matrix a by matrix b.
select the correct choice below, and fill in the answer box to complete your choice.
(type an integer or simplified fraction for each matrix element)
a. the student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row and column elements of the multiplied matrices. the correct matrix is

b. the student made some sign errors in the resulting matrix. the correct matrix is

c. the student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row elements of the multiplied matrices. the correct matrix is

d. the student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding column elements of the multiplied matrices. the correct matrix is

a: \\(\

$$\begin{pmatrix} 6 & 2 \\\\ -3 & 5 \\end{pmatrix}$$

\\) b: \\(\

$$\begin{pmatrix} -1 & 0 \\\\ 4 & -2 \\end{pmatrix}$$

\\) students answer: \\(\

$$\begin{pmatrix} -6 & 0 \\\\ -12 & -10 \\end{pmatrix}$$

\\) (marked with x)

Explanation:

Step1: Recall Matrix Multiplication Rule

For matrix multiplication, if \( A = \begin{pmatrix} a_{11}&a_{12}\\a_{21}&a_{22} \end{pmatrix} \) and \( B=\begin{pmatrix} b_{11}&b_{12}\\b_{21}&b_{22} \end{pmatrix} \), then \( AB=\begin{pmatrix} a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}\\a_{21}b_{11}+a_{22}b_{21}&a_{21}b_{12}+a_{22}b_{22} \end{pmatrix} \).

Given \( A=\begin{pmatrix} 6&2\\-3&5 \end{pmatrix} \), \( B=\begin{pmatrix} -1&0\\4&-2 \end{pmatrix} \).

Step2: Calculate First Element (Row 1, Col 1)

\( (6)(-1)+(2)(4)= -6 + 8 = 2 \). The student got \(-6\), wrong.

Step3: Calculate Second Element (Row 1, Col 2)

\( (6)(0)+(2)(-2)=0 - 4=-4 \). The student got \(0\), wrong.

Step4: Calculate Third Element (Row 2, Col 1)

\( (-3)(-1)+(5)(4)=3 + 20 = 23 \). The student got \(0\), wrong.

Step5: Calculate Fourth Element (Row 2, Col 2)

\( (-3)(0)+(5)(-2)=0 - 10=-10 \). The student got \(-10\) (correct here).

Now, check the options:
- Option A: Says multiplied corresponding elements (which is wrong, matrix multiplication is not element - wise).
- Option B: Says sign errors. Let's see the correct calculations have positive and negative values. The student's errors include sign and wrong operation (element - wise instead of row - column product). But the main error type here: The student multiplied corresponding elements (element - wise) instead of using matrix multiplication (row - column product). Wait, re - check options:
Wait, the options:
A: multiplied corresponding elements (element - wise) instead of matrix multiplication (row - column product).
B: sign errors.
C: multiplied corresponding row elements (no, matrix multiplication is row of first and column of second).
D: multiplied corresponding column elements (no).

Wait, the student's work: \( \begin{pmatrix}6&2\\-3&5\end{pmatrix}\begin{pmatrix}-1&0\\4&-2\end{pmatrix}=\begin{pmatrix}-6&0\\0&-10\end{pmatrix} \). This looks like element - wise multiplication: \( 6\times(-1)=-6 \), \( 2\times0 = 0 \), \( -3\times0 = 0 \), \( 5\times(-2)=-10 \). So the student used element - wise (corresponding elements) multiplication instead of matrix multiplication (row - column product). So Option A is wrong in its description? Wait, no, Option A says "The student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row and column elements of the multiplied matrices." Wait, maybe I misread. Wait, the student's answer is element - wise, but matrix multiplication is row - column product (sum of products of corresponding row and column elements). So the error is that the student did element - wise (corresponding elements) instead of row - column product (matrix multiplication). But let's check the options again:

Wait the options:

A. The student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row and column elements of the multiplied matrices. The correct matrix is \(\begin{pmatrix}2&-4\\23&-10\end{pmatrix}\) (calculated above).

B. The student made some sign errors in the resulting matrix. The correct matrix is... (but our calculation shows more than sign errors, like 6*(-1) is -6 (correct sign), but 2*4 is 8, student didn't add, so it's not just sign).

C. The student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row elements of the multiplied matrices. (No, matrix multiplication is row of first and column of second, not row - row).

D. The student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding column elements of the multiplied matrices. (No, it's row of first and column of second).

So the correct option is B? Wait no, our calculation for row 1, col 1: \(6\times(-1)+2\times4=-6 + 8 = 2\) (student has -6, sign? No, 6*(-1) is -6, but 2*4 is 8, student didn't add, so it's not a sign error. Wait, maybe the original matrices were different? Wait the user's image: Matrix A is \(\begin{pmatrix}6&2\\-3&5\end{pmatrix}\), Matrix B is \(\begin{pmatrix}-1&0\\4&-2\end{pmatrix}\), student's answer is \(\begin{pmatrix}-6&0\\0&-10\end{pmatrix}\).

So the student did element - wise multiplication: (6*(-1), 2*0; -3*0, 5*(-2)). So the error is that the student multiplied corresponding elements (element - wise) instead of using matrix multiplication (row - column product). So Option A: "The student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row and column elements of the multiplied matrices. The correct matrix is \(\begin{pmatrix}2&-4\\23&-10\end{pmatrix}\)".

Now, let's recalculate the correct matrix:

Row 1, Column 1: \(6\times(-1)+2\times4=-6 + 8 = 2\)

Row 1, Column 2: \(6\times0+2\times(-2)=0 - 4=-4\)

Row 2, Column 1: \(-3\times(-1)+5\times4 = 3+20 = 23\)

Row 2, Column 2: \(-3\times0+5\times(-2)=0 - 10=-10\)

So the correct matrix is \(\begin{pmatrix}2&-4\\23&-10\end{pmatrix}\)

Now, the options:

A: describes the error as element - wise instead of matrix multiplication (row - column product) and asks for correct matrix.

B: sign errors. But our calculation shows that the errors are not just sign (e.g., row 1, col 1: -6 vs 2, which is not a sign error but a wrong operation).

C: multiplied corresponding row elements (no, matrix multiplication is row of first and column of second).

D: multiplied corresponding column elements (no).

So the correct option is A. Now, the correct matrix is \(\begin{pmatrix}2&-4\\23&-10\end{pmatrix}\)

Answer:

A. The student multiplied corresponding elements instead of using matrix multiplication to form a new matrix whose elements are the sums of the products of the corresponding row and column elements of the multiplied matrices. The correct matrix is \(

$$\begin{pmatrix}2&-4\\23&-10\end{pmatrix}$$

\)