Given that A and B are square matrices of order 3 with ∣A∣=3 and ∣B∣=−1.
When multiplying a matrix by a scalar k, the determinant becomes:
∣kM∣=kn×∣M∣
where n is the order of the matrix.
For a matrix of order 3:
∣3AB∣=33×∣AB∣
∣3AB∣=27×∣AB∣
The determinant of a product of matrices is:
∣AB∣=∣A∣×∣B∣
Substituting the given values:
∣AB∣=3×(−1)
∣AB∣=−3
Combining the results:
∣3AB∣=27×∣AB∣
∣3AB∣=27×(−3)
∣3AB∣=−81
Therefore, ∣3AB∣=−81.