SAT SUBJECT TEST MATH LEVEL 2

PART 2

REVIEW OF MAJOR TOPICS

CHAPTER 3

Numbers and Operations


3.3 Matrices

DETERMINANTS AND INVERSES OF SQUARE MATRICES

The determinant of an n by n square matrix is a number. The determinant of the 2 by 2 matrix  is denoted by , which equals ad – bc.

EXAMPLES

1. Write an expression for the determinant of .

By definition, .

2. Solve for x: 

The determinant on the left side is x2 – 8x. Use the calculator to evaluate the determinant on the right as 9. This yields the quadratic equation x2 – 8x – 9 = 0. This can be solved by factoring to get x = 9 or x = –1.

For larger square matrices, use the graphing calculator to calculate the determinant (2nd/MATRIX/MATH/det). A matrix whose determinant is zero is called singular. If the determinant is not zero, the matrix is nonsingular.

The product of square n by n matrices is a square n by n matrix. An identity matrix I is a square matrix consisting of 1’s down the main diagonal and 0’s elsewhere. The product of n by n square matrices I and A is A. In other words, I is a multiplicative identity for matrix multiplication.

A nonsingular square n by n matrix A has a multiplicative inverse, A–1, where A–1A = AA–1 = IA–1 can be found on the graphing calculator by entering MATRIX/NAMES/A, which will return A to the home screen, followed by x–1 and ENTER.

3. If  and solve for X when AX = B.

Matrix multiply both sides on the left by A–1:A–1AX = A–1B. This yields IX = X = A–1B = . The fractional form of the answer can be obtained by keying MATH/ENTER/ENTER.

EXERCISES

1. The determinant of  is

      (A)  p – 6

      (B)  p + 6

      (C)  3p – 2

      (D)  3 – 2p

      (E)  –6 – p

2. Find all values of x for which .

      (A)  ±3.78

      (B)  ±4.47

      (C)  ±5.12

      (D)  ±6.19

      (E)  ±6.97

3. If , then X =

      (A)  

      (B)  

      (C)  

      (D)  

      (E)  undefined