Linear Algebra

Exercise T20

page 966

T20 Suppose that A is a square matrix of size n and a C is a scalar. Prove that \(det (aA) = a^n det (A)\).

Proof.

Suppose that A is a square matrix of size n. Then for \(1 \leq i \leq n\),

\(det (A) = (1)^{i+1} [A]_{i1} det (A(i|1)) + (1)^{i+2} [A]_{i2} det (A(i|2))+ ... + (1)^{i+n} [A]_{in} det (A(i|n))\)

According to Theorem EMDRO Elementary Matrices Do Row Operations Suppose that A is an mxn matrix, and B is a matrix of the same size that is obtained from A by a single row operation (De nition RO). Then there is an elementary matrix of size m that will convert A to B via matrix multiplication on the left.

  1. If the row operation swaps rows i and j, then \(B = E_{i,j}A\).

  2. If the row operation multiplies row i by a, then \(B = E_i (a)A\).

  3. If the row operation multiplies row i by a and adds the result to row j, then \(B = E_{i,j} (a)A\).

Repeated applications of Theorem EMDRO 2 allow us to write

\(aA=E_1(a)E_2(a)...E_n(a)A\)

And according to Theorem DEMMM Determinants, Elementary Matrices, Matrix Multiplication Suppose that A is a square matrix of size n and E is any elementary matrix of size n. Then

\(det (EA) = det (E) det (A)\)

Also, According to Theorem DEM Determinants of Elementary Matrices, we know that

\(det (Ei (a)) = a\)

So we have \[\begin{equation} \begin{split} det(aA) &= det(E_1(a)E_2(a)...E_n(a)A) \\ &= det(E_1(a))det(E_2(a))...det(E_n(a))detA \\ &=a^ndetA \end{split} \end{equation}\]