Perform the following operations on the matrix \(\begin{bmatrix} 3 & 4 & 45 \\…
2015
Perform the following operations on the matrix \(\begin{bmatrix} 3 & 4 & 45 \\ 7 & 9 & 105 \\ 13 & 2 & 195 \end{bmatrix}\)
(i) Add the third row to the second row
(ii) Subtract the third column from the first column.
The determinant of the resultant matrix is___________.
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Correct answer: 0
Solution:
The two operations performed are: adding the third row to the second row, and replacing the first column by the first column minus the third column. Both of these are row/column replacement operations that do not change the determinant.
Therefore the determinant of the final matrix equals the determinant of the original matrix.
Look at the original matrix columns: the third column is [45, 105, 195], which is 15 times the first column [3, 7, 13]. So the first and third columns are proportional and the columns are linearly dependent.
A matrix with linearly dependent columns has determinant 0.
Hence the determinant of the resultant matrix is 0.