Barron's SAT Subject Test Math Level 2, 10th Edition (2012)

Part 2. REVIEW OF MAJOR TOPICS

Chapter 3. Numbers and Operations

3.3 Matrices

ADDITION, SUBTRACTION, AND SCALAR
MULTIPLICATION

A matrix is a rectangular array of numbers. The size of a matrix is r by c , where r is the number of rows and c is the number of columns. The numbers in a matrix are called entries, and the entry in the i th row and j th column is named x_ij. Two matrices are equal if they are the same size and their corresponding entries are equal.

EXAMPLES

1. Evaluate x and y if .

These two matrices are equal if y – 3 = 5 and x = 2x + 2. Therefore, y = 8 and x = –2.

If r = 1, the matrix is called a row matrix. If c = 1, the matrix is called a column matrix. If r = c , the matrix is called a square matrix. The numbers from the upper left corner to the bottom right corner of a square matrix form the main diagonal.

Scalar multiplication takes place when each number in a matrix is multiplied by a constant. If two matrices are the same size, they can be added or subtracted by adding or subtracting corresponding entries.

2. Simplify:

3. Solve the matrix equation:

EXERCISES

1. . Find the value of K + J + M.

 (A) 2

 (B) 4

 (C) 6

 (D) 7

 (E) 8

2. Evaluate x and y if .

 (A) x = 0; y = 2

 (B) x = 1; y = 2

 (C) x = −1, 1;

 (D)

 (E) x = 0, ; y = 2

3. Solve for x: .

 (A)

 (B)

 (C)

 (D)

 (E)

MATRIX MULTIPLICATION

Matrix multiplication takes place when two matrices, A and B, are multiplied to form a new matrix, AB. Matrix multiplication is possible only under certain conditions. Suppose A is r₁ by c₁ and B is r₂ by c₂. If c₁ = r₂, then AB is defined and has size r₁ by c₂. The entry x_ij of AB is the i th row of A times the j th column of B. If A and B are square matrices, BA is also defined but not generally equal to AB.

EXAMPLES

1. Evaluate AB = . A is 3 by 2, and B is 2 by 2.

The product matrix is 3 by 2, with entries x_ij calculated as follows:

x₁₁ = (3)(1) + (–1)(5) = –2, the “product” of row 1 of A and column 1 of B

x₁₂ = (3)(–2) + (–1)(–3) = –3, the “product” of row 1 of A and column 2 of B

x₂₁ = (3)(1) + (5)(5) = 28, the “product” of row 2 of A and column 1 of B

x₂₂ = (3)(–2) + (5)(–3) = –21, the “product” of row 2 of A and column 2 of B

x₃₁ = (–2)(1) + (1)(5) = 3, the “product” of row 3 of A and column 1 of B

x₃₂ = (–2)(–2) + (1)(–3) = 1, the “product” of row 3 of A and column 2 of B

Therefore, AB = . Note that BA is not defined.

Matrix calculations can be done on a graphing calculator. To define a matrix, enter 2nd MATRIX, highlight and enter EDIT, enter the number of rows followed by the number of columns, and finally enter the entries. The figure below shows the result of these steps for matrices A and B of Example 1.

To find the product, enter 2nd MATRIX/NAMES/[A], which returns [A] to the home screen. Also enter 2nd MATRIX/NAMES/[B], which returns [B] to the home screen. Hit ENTER again to get the product.

Square matrices of the same size can always be multiplied. However, matrix multiplication is not commutative.

2. If , evaluate AB and BA.

EXERCISES

Use A = [-2 -1 5 9] and B = for questions 1 and 2.

1. The product AB =

 (A)

 (B) [–37 21]

 (C)

 (D)

 (E) product is not defined

2. The first row, second column of the product is

 (A) –5x – 3

 (B) –x – 3

 (C) 1 – x²

 (D) 4x

 (E) 2x + 2

3. If , and AX = B , then the size of X is

 (A) 3 rows, 3 columns

 (B) 3 rows, 2 columns

 (C) 2 rows, 2 columns

 (D) 2 rows, 3 columns

 (E) cannot be determined

4. The chart below shows the number of small and large packages of a certain brand of cereal that were bought over a three-day period. The price of a small box of this brand is $2.99, and the price of a large box is $3.99. Which of the following matrix expressions represents the income, in dollars, received from the sale of cereal each of the three days?

 (A)

 (B)

 (C)

 (D)

 (E)

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 = 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^–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

SOLVING SYSTEMS OF EQUATIONS

An important application of matrices is writing and solving systems of equations in matrix form.

EXAMPLE

Solve the system

This system can be written as AV = B , where A = is the matrix of the

coefficients, is the matrix of the variables, and is the column matrix representing the right side of the system. Multiplying both sides of this equation on the left by A^–1 yields A^–1AV = A^–1B , which reduces to V = A^–1B. If given a system of equations, enter the matrix of coefficients into a matrix (A ), the column matrix of the right side into a second matrix (B ), and find the product A^–1B to display the solution.

EXERCISES

1. Find the matrix equation that represents the system

 (A)

 (B)

 (C)

 (C)

 (E) This system cannot be represented as a matrix equation.

2. Find .

 (A) (–2 0.5)

 (B)

 (C) (–1 3/4)

 (D)

 (E) (–5 –4/5)

Answers and Explanations

Addition, Subtraction, and Scalar Multiplication

1. (C) First find the sum of the matrices to the left of the equals sign: . Since the first row of the matrix to the right of the equals sign is (3 2), K must be 4. Since (J M ) is the bottom row, J = –2 and M = 4. Therefore, K+ J + M = 6.

2. (E) In order for these matrices to be equal, .

  
Therefore, x = 2x² and y = 6y – 10. Solving the first equation yields x = 0, and y = 2.

3. (B) To solve for X , first subtract from both sides of the equation.

  Then .

Matrix Multiplication

1. (B) By definition, AB = = [−37 21]

2. (C) By definition, the first row, second column of the product is (x )(–x ) + (1)(1) = –x² + 1.

3. (B) X must have as many rows as A has columns, which is 3. X must have as many columns as B does, which is 2.

4. (B) Matrix multiplication is row by column. Since the answer must be a 3 by 1 matrix, the only possible answer choice is B.

Determinants and Inverses of Square Matrices

1. (B) By definition, the determinant of is (p )(1) – (3)(–2) = p + 6.

2. (B) Enter the 3 by 3 matrix on the left side of the equation into your graphing calculator and evaluate its determinant (zero). The determinant on the right side of the equation is x² – 20. Therefore x = ± ±4.47.

3. * (C) To find X , multiply both sides of the equation by on the right. Enter both matrices in your calculator, key the product on your graphing calculator, and key MATH/ENTER/ENTER to convert the decimal answer to a fraction.

Solving Systems of Equations

1. (B) First, write the system in standard form: . The matrix form of this equation is .

2. * (D) This is the matrix form AX = B of a system of equations. Multiply both sides of the equation by A^–1 on the left to get the solution X = = A^–1B. Enter the 2 by 2 matrix, A , and the 2 by 1 matrix, B , into your graphing calculator. Return to the home screen and enter A^–1B = .