## 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 *x*^{2} – 8*x*. Use the calculator to evaluate the determinant on the right as 9. This yields the quadratic equation *x*^{2} – 8*x* – 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*^{–1}*A* = *AA*^{–1} = *I*. *A*^{–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*^{–1}*AX* = *A*^{–1}*B*. This yields *IX* = *X* = *A*^{–1}*B* = . 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) 3*p* – 2

(D) 3 – 2*p*

(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