Solved: In Exercises 714, use determinants to decide whether the given matrix is

Section 4.3 EIGENVALUES OF SPECIAL MATRICES Example: Find the eigenvalues.  A=  1 14λ  AI λ    21 −λ  1−4λ det(A−λ ) = 0 21 −λ () ()−8λ 2 12−+λλ 8 0 λλ−270 −±b − 2 4( )( ) λ = 2a ) 7( ) 1 (24±− − λ = 2 ± =± ==22±4 2 11 22 Theorem: MATRICES WITH REAL NUMBER ENTRIES MAY HAVE COMPLEX EIGENVALUES. Example: Let 12 A =22 Find the eigenvalues. Solution: Set det() − λI = 0 and solve for λ. 2λ AI λ −22 λ  1−2λ −2−2 λ =(1 λλ+=2 ) 4 0 =23+ +λ 42 =λλ 36 3± 9-4(1)(65 1±- λ= = 2 2 53±i = 2 For this class, make sure you know that −= i . This is an "imaginary" number. Complex numbers are in the form a+bi. a is the real part and bi is the imaginary part. Several questions will ask for ___+___i and you’ll have to fill in the blanks. Theorem: FOR A REAL MATRIX, COMPLEX EIGENVALUES OCCUR IN CONJUGATE PAIRS; THAT IS, IF a + bi IS AN EIGENVALUE THEN SO IS a - bi . Example: Which of the following COULD be the eigenvalues for a 33× matrix A. 3,i,− 2 B. −+, 1, i C. −−, 2 +i i 3, 4 D. −−, 2 +ii 3,