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Unformatted text preview: CHAPTER 6 MATRIX ALGEBRA A MATRIX is a rectangular array of numbers: A = 1 4 6 7 0 9 8 0 1 3 7 6 A is a (4 x 3) matrix => size = (#rows x # columns) 1 2 3 4 1 2 3 ajk = entry in the jth row and kth column a23 = 9 a41 = 3 A VECTOR is a matrix with only one row or one column: X = Y = 3 0 5  4 Row vector 7 9 2 Column vector MATRIX ADDITION 2 4 6 2 1 3 5 8 + = 4 8 7 5 3 6 0 2 4+2 8+4 0+6 7+0 5+231 6+3 0+5 28 = 6 12 6 7 7 2 9 5 10 SCALAR MULTIPLICATION 4 C = 5 9 6 0 3 51 4 4 5 4 9 4 6 4 4 3 4 5 4 (1) 4 4 = 20 36 240 12204 16 EG1) Let A = 4 6 5 2 4 and B = 3 5 11 64 2 a.) FIND 3A  4B b.) Find a matrix C such that 3C + 2A = B EG2) Let A = c 41 d B = 3 5 11 6 If 3A + 2B = C, what are the values of c and d? and C = 9 2 19 ROW BY COLUMN MULTIPLICATION = 6 9 +8 6 + 3 7 = 123 6 8 3 9 6 7 NOTE: The number of entries in the row on the left must EQUAL the number of entries in the column on the right. IN OTHER WORDS…THE NUMBER OF COLUMNS ON THE LEFT MUST EQUAL THE NUMBER OF ROWS ON THE RIGHT!! MATRIX MULTIPLICATION A B = AB ( m x k ) ( k x n ) = ( m x n ) For A•B to be defined, the # of columns in A must equal the number of rows in B! MATRIX MULTIPLICATION = C ( 2 x 3 ) C ( 3 x 2 ) = ( 2 x 2 ) 3 8 2 6 0 4 4 9 5 0 6 3 = C 3 8 2 6 0 4 4 9 5 0 6 3 1 2 1 2 #12 #12 = (1st row of A) (2nd column of B) = 3 9 + 8 0 + 2 N 3 = 33 EG3) A = 5 6 2 4 6 B = 1 9 3 5 C = 5 6 3 8 2 4 0 4 D = 1 0 0 1 0 0 1 If defined, find the following: a.) BC b.) CB <= NOTE: In general, c.) ABC AB BA d.) B2 e.) DA Click to edit Master subtitle style The Identity Matrix (I): I (n x n) = 1 1 ... ... 1 ... 1 RULE: A!I = I A = A n rows n columns EG4) A = 5 6 2 4 6 B = 1 9 3 5 C = 5 6 3 8 2 4 0 4 For each of the following to be defined, what are the dimensions of I ? a.) CI b.) IB c.) IA EG5) Let A = 5 6 2 4 and C = m 9 3 r For what values of m and r does A·B = C ? B = 3 9 6 6 EG6) Let A = 3 2 5 4 and B = x y For what values of x and y does A·X = B X = 4 6 RULE: Any system of linear equations can be written in the form AX = B, where A is the coefficient matrix for the system, X is the vector of variables, and B is the vector of constants in the equations. This form of the system is called the Matrix Form of the system. EG7) Write the following system of equations in Matrix Form: 3x  2y + z = 8 2x + 3z = 3 6y  2z = Click to edit Master subtitle style PROBLEM : GIVEN A SYSTEM OF EQUATIONS, AX = B, HOW CAN WE SOLVE IT FOR X ? SOLUTION : REASON BY ANALOGY WITH NORMAL ALGEBRA...THAT IS, HOW WOULD YOU SOLVE ax = b FOR x ? EG8) Solve the following system of equations for x and y, 3x + 2y = 4 5x + 4y = 6 … without using the techniques of Chapter 5 ! SECTION 6.2 MATRIX INVERSES Click to edit Master subtitle style A1 = the Inverse of A RULE: A A1 = A1 A = I (A1 acts like the “reciprocal” of the matrix A when multiplying ) NOTE: Only n x n matrices have inverses (and not all n x n matrices have inverses)! Click to edit Master subtitle style How do I find A1 ?...
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This note was uploaded on 11/16/2010 for the course MATH 118 taught by Professor Stevemckinley during the Spring '10 term at Indiana.
 Spring '10
 SteveMcKinley
 Algebra

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