Step 1

Note: Here multiple questions posted, answer to the first three subparts are given. If there is a need for remaining too, kindly post it again with the remark.

Let B be a \(4 \times 4\) matrix,

Let, \(B=\begin{bmatrix}b_{11} & b_{12}&b_{13}&b_{14} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

where, (\(b_{ij}\)) represents the element at the position, i'th row, and j'th column. \(R_i\) represents i'th row and \(C_j\) represents j'th column.

Step 2

1) Double column 1:

\(C_1 \rightarrow 2C_1\)

\(B\sim\begin{bmatrix}2b_{11} & b_{12}&b_{13}&b_{14} \\ 2b_{21} & b_{22}&b_{23}&b_{24} \\ 2b_{31} & b_{32}&b_{33}&b_{34} \\ 2b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

2) Halve row 3:

\(R_3 \rightarrow \frac{1}{2}R_3\)

\(B=\begin{bmatrix}b_{11} & b_{12}&b_{13}&b_{14} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ \frac{1}{2}b_{31} & \frac{1}{2}b_{32}&\frac{1}{2}b_{33}&\frac{1}{2}b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

Step 3

3) Add row 3 to row 1:

\(R_1 \rightarrow R_1+R_3\)

\(B=\begin{bmatrix}b_{11}+b_{31} & b_{12}+b_{32}&b_{13}+b_{33}&b_{14}+b_{34} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

Note: Here multiple questions posted, answer to the first three subparts are given. If there is a need for remaining too, kindly post it again with the remark.

Let B be a \(4 \times 4\) matrix,

Let, \(B=\begin{bmatrix}b_{11} & b_{12}&b_{13}&b_{14} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

where, (\(b_{ij}\)) represents the element at the position, i'th row, and j'th column. \(R_i\) represents i'th row and \(C_j\) represents j'th column.

Step 2

1) Double column 1:

\(C_1 \rightarrow 2C_1\)

\(B\sim\begin{bmatrix}2b_{11} & b_{12}&b_{13}&b_{14} \\ 2b_{21} & b_{22}&b_{23}&b_{24} \\ 2b_{31} & b_{32}&b_{33}&b_{34} \\ 2b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

2) Halve row 3:

\(R_3 \rightarrow \frac{1}{2}R_3\)

\(B=\begin{bmatrix}b_{11} & b_{12}&b_{13}&b_{14} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ \frac{1}{2}b_{31} & \frac{1}{2}b_{32}&\frac{1}{2}b_{33}&\frac{1}{2}b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)

Step 3

3) Add row 3 to row 1:

\(R_1 \rightarrow R_1+R_3\)

\(B=\begin{bmatrix}b_{11}+b_{31} & b_{12}+b_{32}&b_{13}+b_{33}&b_{14}+b_{34} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)