Matrices & Determinants
Determinant of 3x3
Grade Class 12
Question:
If A = <table><tr><td>e<sup>t</sup></td><td>e<sup>-t</sup> cos t</td><td>e<sup>-t</sup> sin t</td></tr><tr><td>e<sup>t</sup></td><td>-e<sup>-t</sup> cos t - e<sup>-t</sup> sin t</td><td>-e<sup>-t</sup> sin t + e<sup>-t</sup> cos t</td></tr><tr><td>e<sup>t</sup></td><td>2e<sup>-t</sup> sin t</td><td>-2e<sup>-t</sup> cos t</td></tr></table> Then A is -
(1) Invertible only if t = π/2
(2) not invertible for any t ∈ R
(3) invertible for all t ∈ R
(4) invertible only if t = π
Step-by-Step Solution
Key Concept: Calculate the determinant of matrix A. If det(A) is non-zero for all t, then A is invertible for all t. If det(A) is zero for some t, check the condition.
The determinant of the given matrix A can be calculated by expanding along the first row. After simplification, it is found that det(A) = e<sup>t</sup> * e<sup>-2t</sup> * (constant). Specifically, evaluating the determinant shows it is non-zero for all t, thus A is invertible for all t.
Correct Answer: 1