58. Writing Evaluate the determinant of each matrix. Describe any patterns.
    1. [ 1 2 3 1 2 3 1 2 3 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 2 , column 3 3 , row2 column 1 , 1 , column 2 2 , column 3 3 , row3 column 1 , 1 , column 2 2 , column 3 3 , end matrix
    2. [ − 1 − 2 − 3 − 3 − 2 − 1 − 1 − 2 − 3 ] . matrix with 3 rows and 3 columns , row1 column 1 , negative 1 , column 2 negative 2 , column 3 negative 3 , row2 column 1 , negative 3 , column 2 negative 2 , column 3 negative 1 , row3 column 1 , negative 1 , column 2 negative 2 , column 3 negative 3 , end matrix
    3. [ 1 2 3 2 3 1 1 2 3 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 2 , column 3 3 , row2 column 1 , 2 , column 2 3 , column 3 1 , row3 column 1 , 1 , column 2 2 , column 3 3 , end matrix
    4. [ − 1 2 − 3 2 − 3 − 1 − 1 2 − 3 ] . matrix with 3 rows and 3 columns , row1 column 1 , negative 1 , column 2 2 , column 3 negative 3 , row2 column 1 , 2 , column 2 negative 3 , column 3 negative 1 , row3 column 1 , negative 1 , column 2 2 , column 3 negative 3 , end matrix
59. Reasoning For what value of x will matrix A have no inverse? A = [ 1 2 3 x ] eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 2 , row2 column 1 , 3 , column 2 x , end matrix

C Challenge

60. Reasoning Suppose A = [ a b c d ] , eh equals . matrix with 2 rows and 2 columns , row1 column 1 , eh , column 2 b , row2 column 1 , c , column 2 d , end matrix . comma  For what values of a, b, c, and d will A be its own inverse? (Hint: There is more than one correct answer.)
61. Let M = [ a b c d ] m equals . matrix with 2 rows and 2 columns , row1 column 1 , eh , column 2 b , row2 column 1 , c , column 2 d , end matrix  and N = [ e f g h ] . n equals . matrix with 2 rows and 2 columns , row1 column 1 , e , column 2 f , row2 column 1 , g , column 2 h , end matrix . .  Prove that the product of the determinants of M and N equals the determinant of the matrix product MN.

Standardized Test Prep

GRIDDED RESPONSE

SAT/ACT

62. What is the determinant of [ − 2 − 3 5 0 ] ? . matrix with 2 rows and 2 columns , row1 column 1 , negative 2 , column 2 negative 3 , row2 column 1 , 5 , column 2 0 , end matrix . question mark
63. If A = [ 4 2 − 3 − 1 ] , eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 4 , column 2 2 , row2 column 1 , negative 3 , column 2 negative 1 , end matrix . comma  and the inverse of A is x [ − 1 − 2 3 4 ] , x . matrix with 2 rows and 2 columns , row1 column 1 , negative 1 , column 2 negative 2 , row2 column 1 , 3 , column 2 4 , end matrix . comma  what is the value of x? Enter your answer as a fraction.
64. What is the value of 6 ! 8 ! ? fraction 6 factorial , over 8 factorial end fraction . question mark  Give your answer as a fraction in simplest terms.
65. If log ( 7 y − 5 ) = 2 , log open 7 y minus 5 close equals 2 comma  what is the value of y?
66. If the equation for a circle is x 2 + y 2 − 2 x + 6 y − 6 = 0 , x squared , plus , y squared , minus 2 x plus 6 y minus 6 equals 0 comma  what is its radius?

Mixed Review

See Lesson 12-2.

Solve each matrix equation.

67. 2 [ − 1 3 − 2 0 ]   −   3 X = [ − 8 − 9 − 7 − 3 ] 2 . matrix with 2 rows and 2 columns , row1 column 1 , negative 1 , column 2 3 , row2 column 1 , negative 2 , column 2 0 , end matrix . minus 3 x equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 8 , column 2 negative 9 , row2 column 1 , negative 7 , column 2 negative 3 , end matrix
68. 2 X + 3 [ 4 − 6 8 − 3 ]   = [ − 8 20 − 16 5 ] 2 x plus 3 . matrix with 2 rows and 2 columns , row1 column 1 , 4 , column 2 negative 6 , row2 column 1 , 8 , column 2 negative 3 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 8 , column 2 20 , row2 column 1 , negative 16 , column 2 5 , end matrix

See Lesson 11-1.

Evaluate each expression.

69. 6!
70. 9!
71. 15 ! 5 ! fraction 15 factorial , over 5 factorial end fraction
72. 12 ! 6 ! 3 ! fraction 12 factorial , over 6 factorial 3 factorial end fraction

Get Ready! To prepare for Lesson 12-4, do Exercises 73–76.

See Lesson 3-5.

Solve each system.

73. { x = 5 x − y + z = 5 x + y − z = − 5 left brace . table with 3 rows and 1 column , row1 column 1 , x equals 5 , row2 column 1 , x minus y plus z equals 5 , row3 column 1 , x plus y minus z equals negative 5 , end table
74. { x − y − z = 9 3 x + 2 z = 12 x = y − 2 z left brace . table with 3 rows and 1 column , row1 column 1 , x minus y minus z equals 9 , row2 column 1 , 3 x plus 2 z equals 12 , row3 column 1 , x equals y minus 2 z , end table
75. { − x + 2 y + z = 0 y = − 2 x + 3 z = 3 x left brace . table with 3 rows and 1 column , row1 column 1 , negative x plus 2 y plus z equals 0 , row2 column 1 , y equals negative 2 x plus 3 , row3 column 1 , z equals 3 x , end table
76. { 5 x − 4 y − 3 z = 3 z = y + x x = 3 y + 1 left brace . table with 3 rows and 1 column , row1 column 1 , 5 x minus 4 y minus 3 z equals 3 , row2 column 1 , z equals y plus x , row3 column 1 , x equals 3 y plus 1 , end table

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