Matrix Operations

Matrices appear with limited frequency on the ACT. (I have yet to see them used on the SAT.) There are a few things that you may be asked to understand about matrices - these are listed below.

1. Adding/subtracting matrices
2. Multiplying matrices
3. Multiplying/dividing matrix elements by a constant or variable
4. Finding the determinant of a matrix

In order for a matrix addition/subtraction to take place, the matrices in question must have identical dimensions. This means that all matrices must have the same number of rows and the same number of columns. (The number of rows and columns do not have to match within a matrix.) We can add the two matrices below.

Each matrix has six rows and three columns, allowing them to be added/subtracted. When we add/subtract, we would add/subtract the entries in the corresponding locations, forming a new matrix of the same dimensions as the original two matrices. Multiplying matrices requires that the number of columns in the first matrix match the number of rows in the second matrix. Since the first matrix above has three columns and the second matrix has six rows, we cannot multiply the above matrices. We, however, can multiply the matrices below.

In order to perform the multiplication, we multiply each column in the second matrix by each row in the first matrix. Each such product of a column and a row produces a single number. (The first number, placed in the first row and the first column, would be -9*19 + -2*18 + 2*-19 = -245.  The second number, placed in the second row and first column, would be -9*-3 + -2*8 + 2*-5 = 1.)  The resulting product matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.  The product matrix above would have six rows and five columns.  The two matrices below cannot be multiplied (columns of the first do not match rows of the second), showing that matrix multiplication is not commutative.

When a single term appears outside of a matrix, you simply multiply all elements within the matrix by that value or term.

The determinant of a matrix determines whether the system defined by the matrix has a unique solution.  If the determinant is non-zero, there is a unique solution to the system.  In order for a matrix to have a determinant, it must be a square matrix - its number of rows must match its number of columns.  Both of the matrices below have determinants.

The determinant of the second matrix is 6*14 - 5*7 = 84 - 35 = 49.  The determinant of the first matrix is 13*(-2)*(-6) + 10*12*(-3) + (-6)*16*12 - (-6)*(-2)*(-3) - 10*16*(-6) - 13*12*12 = 156 + -360 + -1152 - -36 - -960 - 1872 = 156 - 360 - 1152 + 36 + 960 - 1872 = -2232.  A full explanation for how to calculate determinants for larger matrices is beyond what you will need to know for the ACT/SAT.

Revisiting Function Equality

Let us revisit function equality. Consider the simple scenario described below.

• If f(x) = 5x and g(x) = ax, and if f(x) = g(x) for all values of x, then what is the value of a?

This may seem like a trivial example, but it applies to problems of the same general type. So, let us look at the following table.

x	f(x)	g(x)	a
-5	-25	-5a	5
-4	-20	-4a	5
-3	-15	-3a	5
-2	-10	-2a	5
-1	-5	-a	5
0	0	0	anything
1	5	a	5
2	10	2a	5
3	15	3a	5
4	20	4a	5
5	25	5a	5

Only when x = 0 will a take on any value other than 5, but since the two functions must be equal for all values of x, a must be 5. This idea can be expanded to more complicated problems. Let us look at another example.

• If (a + 7)(x² + 3x - 4) = (2x + b)(2x + 8) for all values of x, then what is the value of b?

There are at least two ways to solve this problem. One of them, the one that you are likely to default to, will take much longer than the other. The longer, brute force method has you applying the Distributive Property to both sides and getting the following.

• ax² + 7x² + 3ax + 21x - 4a - 28 = 4x² + 16x + 2bx + 8b

Now, we can collect like terms.

• (a + 7)x² + (3a + 21)x - 4(a + 7) = 4x² + (2b + 16)x + 8b

From the previous example, it should be evident that these two expressions will only be equal if the corresponding coefficients are equal.

• (a + 7)x² = 4x² → a + 7 = 4
• (3a + 21)x = (2b + 16)x → 3a + 21 = 2b + 16
• -4(a + 7) = 8b

From the first of these three equations, we can see that a = -3. We can then use this value in either of the other equations to conclude that b = -2. However, this can be solved with far less work by analyzing the equation more closely early on. Let us rewrite the problem.

• If (a + 7)(x² + 3x - 4) = (2x + b)(2x + 8) for all values of x, then what is the value of b?

If we look at the right side of the equation, we can clearly see that the x² term will be 4x². This then forces the x² term on the left to also be 4x², forcing a + 7 = 4. If a + 7 = 4, then the constant term on the left hand side will have to equal -16, forcing the constant term on the right hand side, 8b, to also equal -16. If 8b = -16, b = -2.

We get to the same answer with either method, but if you practice the second of the two methods, you can become more proficient at getting to the answer faster.

What is a Function?

A function is a set of instructions that you perform on an input or on multiple inputs in order to produce an output. Whenever you are given a function, you should identify what the input is (inputs are) and figure out what instructions you are being asked to perform on that input (those inputs). It is important that you not get hung up on what specific variable is being used (or variables are being used); the more important thing to do is to understand what instructions/operations are being asked to be performed on the input(s). Let's take a look at an example. What is the difference between the following two functions?

• f(x) = x² + 3x
• f(y) = y² + 3y

The answer is that there is no difference between these two functions. In each case, we have a single input and have these instructions.

1. Square your input
2. Add three times your input

The fact that we use a different variable in the two cases is irrelevant. Each function has the same number of inputs, and we perform the same instructions on those inputs. The functions are thus equivalent. Let's look at another example.

• If f(x) = -x² + x + 17, then what is f(x - 2)?

Here are our instructions.

1. Take the negative of the square of your input
2. Add your input
3. Add 17

We are asked to find f(x - 2); our input has changed, but our instructions have not.

• f(x - 2) = -(x - 2)² + (x - 2) + 17
• f(x - 2) = -(x² - 4x + 4) + x - 2 + 17
• f(x - 2) = -x² + 4x - 4 + x + 15
• f(x - 2) = -x² + 5x + 11

Let's look at one more example. It's similar but a bit trickier.

• If f(x - 3) = x² + 3, then what is f(x)?

In this case, our input is x - 3, but we want to turn our input into just x. This may seem complicated, but it really is not. All we have to do is to substitute x + 3 everywhere that we see x. Let's do it.

• f((x + 3) - 3) = (x + 3)² + 3
• f(x + 3 - 3) = (x² + 6x + 9) + 3
• f(x) = x² + 6x + 12

Here are a couple of extra problems for you to try if you so please.

Example Problems

If f(x) = 2x² + 4x - 2, then what is f(x - 3)?

If f(x + 5) = x² + 14x + 53, then what is f(x)?