Matrix Transpose and Determinant?
If A is a 2x2 matrix and AT represents the transpose of matrix A, then what can you infer about the relationship between the determinant of A and the determinant of AT when A is a singular matrix?
1 Answer
📌 CONCEPT: The transpose of a matrix is obtained by interchanging its rows into columns or columns into rows.
📐 RULE / FORMULA: The determinant of the transpose of a matrix is equal to the determinant of the original matrix, i.e., det(AT) = det(A).
💡 WORKED EXAMPLE: Consider a 2x2 matrix A = [[a, b], [c, d]]. The determinant of A is given by det(A) = ad - bc. The transpose of A is AT = [[a, c], [b, d]]. Its determinant is det(AT) = ad - bc, which is the same as the determinant of A. Therefore, det(AT) = det(A).
⚠️ COMMON MISTAKE: Students often confuse the concept of the transpose with that of the adjoint or the inverse of a matrix.
28 Sept 26
🔗 More from Matrices
Practice this chapter
Get AI-generated board exam questions, track your mastery, and identify weak spots.
Start Free →