3.2 Determinants; Mtx Inverses

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3.2 Determinants; Mtx Inverses. Theorem 1- Product Theorem. If A and B are (n x n), then det(AB)=det A det B (come back to prove later) Show true for 2 x 2 of random variables. Extension. Using induction, we could show that: det(A 1 A 2 …A k ) = detA 1 detA 2 …detA k

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### 3.2 Determinants; Mtx Inverses

Theorem 1- Product Theorem
• If A and B are (n x n), then det(AB)=det A det B
• (come back to prove later)
• Show true for 2 x 2 of random variables
Extension
• Using induction, we could show that:
• det(A1A2…Ak) = detA1detA2…detAk
• also: det (Ak) = (det A)k
Theorem 2
• An (n x n) matrix A is invertible iff det A ≠ 0. If it is invertible,
• Proof: ==> given A invertible, AA-1=I
• and
Proof (continued)
• <== Given det A ≠ 0
• A can clearly be taken to reduced row ech form w/ no row of zeros (call it R = Ek…E2E1A) (otherwise the determinant would be 0)
• (det R = det Ek … det E2 det E1 det A≠0)
• Since R has no row of zeros, R=I, and A is clearly invertible.
Example
• Find all values of b for which A will have an inverse.
Theorem 3
• If A is any square matrix, det AT = det A
• Proof: For E of type I or type II, ET = E (show ex)
• For E of type III, ET is also of type III, and det E = 1 = det ET by theorem 2 in 3.1 (which says that if we add a multiple of a row to another row, we do not change the determinant).
• So det E = det ET for all E
• Given A is any square matrix:
• If A is not invertible, neither is AT (since the row operations to reduce A which would take A to a row of zeros could be used as column ops on AT to get a column of zeros) so det A = 0 = det AT
Theorem 3 (continued)
• If A is invertible, then A = Ek…E2E1 and AT= E1TE2T…EkT
• So det AT = det E1T det E2T …det EkT = detE1detE2…detEk = det A 
Examples
• If det A =3, det B =-2 find det (A-1B4AT)
• A square matrix is orthogonal is A-1 = AT. Find det A if A is orthogonal.
• I = AA-1 = AAT
• 1 = det I = det A det AT = (det A)2
• So det A = ± 1
• Adjoint of a (2x2) is just the right part of inverse:
• Recall that:
• Now we will show that it is also true for larger square matrices.
• If A is square, the cofactor matrix of A , [Cij(A)], is the matrix whose (I,j) entry is the (i,j) cofactor of A.
• The adjoint of A, adj(A), is the transpose of the cofactor matrix:
• Now we need to show that this will allow our definition of an inverse to hold true for all square matrices:
Example
• Find the adjoint of A:
• So we could find det(A)
For (nxn)
• A(adjA) = (detA)I for any (nxn): ex. (3 x 3)
• we have 0’s off diag since they are like determinants of matrices with two identical rows (like prop 5 of last chapter)
• If A is any square matrix, then
• If det A ≠ 0,
• Good theory, but not a great way to find A-1
Example
• Use thm 4 to find the values of c which make A invertible:

c ≠ 0

Linear Equations
• Recall that if AX = B, and if A is invertible (det A ≠ 0), then
• X = A-1B So...
Finding determinants is easier
• the right part is just the det of a matrix formed by replacing column i with the B column matrix
Theorem 5: Cramer’s Rule

If A is an invertible (n x n) matrix, the solution to the system AX = B of n equations in n variables is:

Where Ai is the matrix obtained by replacing column i of A with the column matrix B.

This is not very practical for large matrices, and it does not give a solution when A is not invertible

Examples
• Solve the following using Cramer’s rule.
Proof of Theorem 1
• for A,B (nxn): det AB = det A detB
• det E = -1 if E is type 1.
• = u if E is type 2 (and u is multiplied by one row of I)
• = 1 if E is of type 3
• If E is applied to A, we get EA
• det (EA) = -det(A) if E is of type I
• = udet(A) if E is of type II
• = det (A) if E is of type III
• So det (EA) = det E det A
Continued...
• So if we apply more elementary matrices:
• det(E2E1A) = det E2(det(E1A)) = det E2 det E1 det A
• This could continue and we get the following:
• Lemma 1: If E1, E2, …, Ek are (n x n) elementary matrices,
• and A is (n x n), then:
• det(Ek …E2 E1A) = det Ek … det E2 det E1 det A
Continued
• Lemma 2: If A is a noninvertible square matrix, then det A =0
• Proof: A is not invertible ==> when we put A into reduced row echelon form, the resulting matrix, R will have a row of zeros.
• det R = 0
• det R = det (Ek…E2E1A) = det Ek … det E2 det E1 det A= 0
• since det E’s never 0, det A = 0
Proving Theorem 1 (finally)
• Show that det AB = det A det B
• Proof : Case 1: A is not invertible:
• Then det A = 0
• If AB were invertible, then AB(AB)-1 = I
• so A(BB-1A) = I, which would mean that A is
• invertible, but it is not, so AB is not invertible.
• Therefore, det AB = 0 = det A det B
Continued..
• Case 2: A is invertible:
• A is a product of elementary matrices so
• det A = det(Ek…E2E1) = det Ek …det E2 det E1 (by L 1)
• det(AB) = det (Ek … E2 E1 B)
• = det Ek … det E2 det E1 det B (by L 1)
• = det A det B 