Question

matrix multiplication: specific entries
for each problem below, multiply the given 3 × 3 matrix by the 3 × 4 matrix. do not compute the entire product. find only the following entries of the resulting 3 × 4 matrix:
1.
\\( m = \

$$\begin{pmatrix} 2 & -1 & 4 \\\\ 0 & 3 & 5 \\\\ 1 & 2 & -1 \\end{pmatrix}$$

\\), \\( n = \

$$\begin{pmatrix} 1 & 2 & 0 & 3 \\\\ -4 & 5 & 1 & 2 \\\\ -6 & -1 & 4 & 0 \\end{pmatrix}$$

\\)
find the (2, 3) element of \\( mn \\).
2.
\\( p = \

$$\begin{pmatrix} -3 & 2 & 1 \\\\ 4 & 0 & -2 \\\\ -2 & -3 & 4 \\end{pmatrix}$$

\\), \\( q = \

$$\begin{pmatrix} 5 & -1 & 3 & 2 \\\\ 7 & 4 & -2 & 1 \\\\ -6 & 0 & 5 & -3 \\end{pmatrix}$$

\\)
find the (3, 4) element of \\( pq \\).

Explanation:

Problem 1:

Step1: Recall matrix multiplication rule

To find the \((i, j)\) entry of \(MN\), we use the dot product of the \(i\)-th row of \(M\) and the \(j\)-th column of \(N\). For the \((2, 3)\) entry, we take the 2nd row of \(M\) and 3rd column of \(N\).
The 2nd row of \(M\) is \([0, 3, 5]\), and the 3rd column of \(N\) is \([0, 1, 4]\).

Step2: Compute the dot product

The dot product is \(0\times0 + 3\times1 + 5\times4\).
Calculating each term: \(0\times0 = 0\), \(3\times1 = 3\), \(5\times4 = 20\).
Summing them: \(0 + 3 + 20 = 23\).

Step1: Recall matrix multiplication rule

To find the \((i, j)\) entry of \(PQ\), we use the dot product of the \(i\)-th row of \(P\) and the \(j\)-th column of \(Q\). For the \((3, 4)\) entry, we take the 3rd row of \(P\) and 4th column of \(Q\).
The 3rd row of \(P\) is \([-2, -3, 4]\), and the 4th column of \(Q\) is \([2, 1, -3]\).

Step2: Compute the dot product

The dot product is \(-2\times2 + (-3)\times1 + 4\times(-3)\).
Calculating each term: \(-2\times2 = -4\), \(-3\times1 = -3\), \(4\times(-3) = -12\).
Summing them: \(-4 + (-3) + (-12) = -19\).

Answer:

23

Problem 2: