Let B be a 4 times 4 matrix to which we apply the following operations: 1. double column 1, 2. halve row 3, 3. add row 3 to row 1, 4. interchange columns 1 and 4, 5. subtract row 2 from each of the other rows, 6. replace column 4 by column 3, 7. delete column 1 (so that the column dimension is reduced by 1). (a) Write the result as a product of eight matrices. (b) Write it again as a product ABC (same B) of three matrices.

Question
Matrices
asked 2021-02-25
Let B be a \(4 \times 4\) matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (so that the column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product ABC (same B) of three matrices.

Answers (1)

2021-02-26
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:
\(R_1 \rightarrow R_1+R_3\)
\(B=\begin{bmatrix}b_{11}+b_{31} & b_{12}+b_{32}&b_{13}+b_{33}&b_{14}+b_{34} \\ b_{21} & b_{22}&b_{23}&b_{24} \\ b_{31} & b_{32}&b_{33}&b_{34} \\\ b_{41} & b_{42}&b_{43}&b_{44} \end{bmatrix}\)
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