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# How to find determinant of a 4th order matrix

askIITians Faculty 27 Points
7 years ago
Any matrixAand its transpose have the same determinant, meaning

2.
The determinant of a triangular matrix is the product of the entries on the diagonal.
3.
If we interchange two rows, the determinant of the new matrix is the opposite of the old one.
4.
If we multiply one row with a constant, the determinant of the new matrix is the determinant of the old one multiplied by the constant.
5.
If we add one row to another one multiplied by a constant, the determinant of the new matrix is the same as the old one.
6.
We have

In particular, ifAis invertible (which happens if and only if), then

So let us see how this works in case of a matrix of order 4.

Example.Evaluate

We have

If we subtract every row multiplied by the appropriate number from the first row, we get

We do not touch the first row and work with the other rows. We interchange the second with the third to get

If we subtract every row multiplied by the appropriate number from the second row, we get

Using previous properties, we have

If we multiply the third row by 13 and add it to the fourth, we get

which is equal to 3. Putting all the numbers together, we get

Thank YouRuchiAskiitians Faculty