1. Given $$ M=\left[\begin{array}{cccc}1 & 3 & -5 & 9 \ -2 & 1 & 8 & -1 \ 5 & 2 & -4 & -1 \ 2 & -4 & 7 & 6\end{array} ight] $$ and $$ N=\left[\begin{array}{ccc}1 & -7 & -1 \ 5 & 3 & 6 \ -3 & 1 & 0 \ 2 & 4 & -3\end{array} ight] $$. (a) Without calculating the product $$ M N $$ determine (i) the entry in row 4 , column 3 of $$ M N $$, (Ans: -44$$ ) $$ (ii) the entries in column 2 of $$ M N $$, and $$ \left[\begin{array}{c}33 \ 21 \ -37 \ 5\end{array} ight] $$ (Ans: (iii) the entries in row 3 of $$ M N . \quad $$ (Ans: $$ \left[\begin{array}{ccc}25 & -37 & 10\end{array} ight] $$ ) (b) Obtain the matrix $$ N^{T} M . \quad $$ (Ans: $$ \left.\left[\begin{array}{cccc}-20 & -6 & 61 & 19 \ 0 & -32 & 83 & -43 \ -19 & 15 & 32 & -33\end{array} ight] ight) $$

Question

1. Given $$ M=\left[\begin{array}{cccc}1 & 3 & -5 & 9 \ -2 & 1 & 8 & -1 \ 5 & 2 & -4 & -1 \ 2 & -4 & 7 & 6\end{array}ight] $$ and $$ N=\left[\begin{array}{ccc}1 & -7 & -1 \ 5 & 3 & 6 \ -3 & 1 & 0 \ 2 & 4 & -3\end{array}ight] $$. (a) Without calculating the product $$ M N $$ determine (i) the entry in row 4 , column 3 of $$ M N $$, (Ans: -44$$ ) $$ (ii) the entries in column 2 of $$ M N $$, and $$ \left[\begin{array}{c}33 \ 21 \ -37 \ 5\end{array}ight] $$ (Ans: (iii) the entries in row 3 of $$ M N . \quad $$ (Ans: $$ \left[\begin{array}{ccc}25 & -37 & 10\end{array}ight] $$ ) (b) Obtain the matrix $$ N^{T} M . \quad $$ (Ans: $$ \left.\left[\begin{array}{cccc}-20 & -6 & 61 & 19 \ 0 & -32 & 83 & -43 \ -19 & 15 & 32 & -33\end{array}ight]ight) $$

UpStudy AI · Accepted Answer

To solve the problem, we will analyze the matrices $$ M $$ and $$ N $$ and use properties of matrix multiplication to find the required entries without directly calculating the entire product $$ MN $$.

### Part (a)

#### (i) Entry in row 4, column 3 of $$ MN $$

The entry in row $$ i $$ and column $$ j $$ of the product $$ MN $$ is calculated as the dot product of the $$ i $$-th row of $$ M $$ and the $$ j $$-th column of $$ N $$.

For row 4 of $$ M $$ and column 3 of $$ N $$:
- Row 4 of $$ M $$: $$ [2, -4, 7, 6] $$
- Column 3 of $$ N $$: $$ [-1, 6, 0, -3] $$

Calculating the dot product:
$$
2 \cdot (-1) + (-4) \cdot 6 + 7 \cdot 0 + 6 \cdot (-3) = -2 - 24 + 0 - 18 = -44
$$

Thus, the entry in row 4, column 3 of $$ MN $$ is $$-44$$.

#### (ii) Entries in column 2 of $$ MN $$

The entries in column 2 of $$ MN $$ are obtained by taking the dot product of each row of $$ M $$ with column 2 of $$ N $$.

Column 2 of $$ N $$: $$ [-7, 3, 1, 4] $$

Calculating each entry:

1. **Row 1 of $$ M $$**:
   $$
   1 \cdot (-7) + 3 \cdot 3 + (-5) \cdot 1 + 9 \cdot 4 = -7 + 9 - 5 + 36 = 33
   $$

2. **Row 2 of $$ M $$**:
   $$
   -2 \cdot (-7) + 1 \cdot 3 + 8 \cdot 1 + (-1) \cdot 4 = 14 + 3 + 8 - 4 = 21
   $$

3. **Row 3 of $$ M $$**:
   $$
   5 \cdot (-7) + 2 \cdot 3 + (-4) \cdot 1 + (-1) \cdot 4 = -35 + 6 - 4 - 4 = -37
   $$

4. **Row 4 of $$ M $$**:
   $$
   2 \cdot (-7) + (-4) \cdot 3 + 7 \cdot 1 + 6 \cdot 4 = -14 - 12 + 7 + 24 = 5
   $$

Thus, the entries in column 2 of $$ MN $$ are:
$$
\begin{bmatrix}
33 \
21 \
-37 \
5
\end{bmatrix}
$$

#### (iii) Entries in row 3 of $$ MN $$

The entries in row 3 of $$ MN $$ are obtained by taking the dot product of row 3 of $$ M $$ with each column of $$ N $$.

Row 3 of $$ M $$: $$ [5, 2, -4, -1] $$

Calculating each entry:

1. **Column 1 of $$ N $$**:
   $$
   5 \cdot 1 + 2 \cdot 5 + (-4) \cdot (-3) + (-1) \cdot 2 = 5 + 10 + 12 - 2 = 25
   $$

2. **Column 2 of $$ N $$**:
   $$
   5 \cdot (-7) + 2 \cdot 3 + (-4) \cdot 1 + (-1) \cdot 4 = -35 + 6 - 4 - 4 = -37
   $$

3. **Column 3 of $$ N $$**:
   $$
   5 \cdot (-1) + 2 \cdot 6 + (-4) \cdot 0 + (-1) \cdot (-3) = -5 + 12 + 0 + 3 = 10
   $$

Thus, the entries in row 3 of $$ MN $$ are:
$$
\begin{bmatrix}
25 & -37 & 10
\end{bmatrix}
$$

### Part (b)

To obtain the matrix $$ N^T M $$, we first need to find the transpose of $$ N $$.

The transpose $$ N^T $$ is:
$$
N^T = \begin{bmatrix}
1 & 5 & -3 & 2 \
-7 & 3 & 1 & 4 \
-1 & 6 & 0 & -3
\end{bmatrix}
$$

Now we will calculate the product $$ N^T M $$.

Calculating each entry of $$ N^T M $$:

1. **Row 1 of $$ N^T $$** with each column of $$ M $$:
   - Column 1 of $$ M $$: $$ [1, -2, 5, 2] $$
   $$
   1 \cdot 1 + 5 \cdot (-2) + (-3) \cdot 5 + 2 \cdot 2 = 1 - 10 - 15 + 4 = -20
   $$
   - Column 2 of $$ M $$: $$ [3, 1, 2, -4] $$
   $$
   1 \cdot 3 + 5 \cdot 1 + (-3) \cdot 2 + 2 \cdot (-4) = 3 + 5 - 6 - 8 = -6
   $$
   - Column 3 of $$ M $$: $$ [-5, 8, -4, 7] $$
   $$
   1 \cdot (-5) + 5 \cdot 8 + (-3) \cdot (-4) + 2 \cdot 7 = -5 + 40 + 12 + 14 = 61
   $$
   - Column 4 of $$ M $$: $$ [9, -1, -1, 6] $$
   $$
   1 \cdot 9 + 5 \cdot (-1) + (-3) \cdot (-1) + 2 \cdot 6 = 9 - 5 + 3 + 12 = 19
   $$

2. **Row 2 of $$ N^T $$** with each column of $$ M $$:
   - Column 1 of $$ M $$: $$ [1, -2, 5, 2] $$
   $$
   -
**Part (a):** 1. **Entry in row 4, column 3 of $$ MN $$:** $$-44$$ 2. **Entries in column 2 of $$ MN $$:** \[ \begin{bmatrix} 33 \ 21 \ -37 \ 5 \end{bmatrix} $$ 3. **Entries in row 3 of $$ MN $$:** $$ \begin{bmatrix} 25 & -37 & 10 \end{bmatrix} $$ **Part (b):** $$ N^T M = \begin{bmatrix} -20 & -6 & 61 & 19 \ 0 & -32 & 83 & -43 \ -19 & 15 & 32 & -33 \end{bmatrix} $$

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