Taking Down the Matrix
By Leo Lam
(Test Preparation (ACT/SAT/SSAT), Math and Physics tutor at The Edge Learning Center)
In my last blog, I talked about the complex number, a topic that is becoming more prominent in both SAT and ACT. Today, I will discuss another advanced math topic you may encounter on these college prep tests: matrix (plural: matrices).
While SAT does not test the concept of the matrix, this topic has become a staple of the ACT math section. And if you are taking the SAT subject test, you will also find questions on matrices in the Math Level 2 test. You will not encounter numerous questions about matrices on either tests, but in order to achieve the highest score possible, you would need to know the fundamental concepts behind this topic.
What is a matrix?
Simply put, a matrix is a collection of values or expressions. Unlike a set, these values are order-specific and arranged in a rectangular array. The numbers of rows and columns determine the size of a matrix. For example:
The matrix A has 2 rows and 3 columns. We call this a 2 X 3(read as “two by three”) matrix. Each value inside a matrix is referred to as an entry or element. A common way to specify an entry inside a matrix is by using the notation, where the subscriptsand represent the row and column, respectively.
Thus, in our matrix,would be 5, whilewould be -11.
In order for two matrices to be considered equivalent, they must have the exact same size and corresponding entries.
What do I need to know?
There are three common concepts that are asked on these tests: 1) operating with matrices, 2) finding the determinant, and 3) translating a problem into a matrix.
Operating with Matrices
Matrices can be added, subtracted, or multiplied (but not divided, we will get into that later). Addition and subtraction are very straightforward: you add/subtract the corresponding entries from the two matrices as long as they are the same size. For example:
This means addition or subtraction between two matrices with different sizes is not possible.
Multiplication works very differently. First, there are two types of multiplications: scalar multiplication and matrix multiplication. Scalar multiplication involves multiplying a matrix with a number. The size of the matrix is irrelevant. What is important is that every single entry inside must be multiplied by the same value. For example:
If you want to divide each entry inside the matrix, you can multiply by the reciprocal of the value instead, i.e. dividing by 3 can be calculated as multiplying by 1/3.
Matrix multiplication, on the other hand, is more complicated. First, there are a few rules you must remember:
- Matrix multiplication is not always commutative, i.e.
- .In order to multiply two matrices together, the number of columns in the first matrix MUST be equal to the number of rows in the second matrix.
- The product has the same number of rows as the first matrix and the same number of columns as the second matrix.
If you follow these rules closely, you will realize that many answer choices in a test can be eliminated because one or more of the rules are broken. Let us take a look at some examples.
Let A be a 2 X 3 matrix, B be a 3 X 4 matrix, and C be a 3 X 2 matrix.
A X B is defined in this case, and the result, by following rule III, is a 2 X 4 matrix. However, B X C is undefined, as it does not follow rule II. Furthermore, both A X C and C X A are defined, but the result of A X C is a 2 X 2 matrix, while the result of C X A is a 3 X 3. Thus, the results are different matrices, which demonstrates rule I.
The actual process of multiplication gets even more complex. In a nutshell, each entry of the resultant matrix is a dot product between a row vector of the first matrix and the column vector of the second matrix (for those of you who have not studied vector, here is a quick lesson on vector and dot product).
Instead of going through the nitty-gritty process, you should rely on your graphing calculator to get the result. Here are two YouTube videos showing you how to use your TI-84 or TI-Nspire to multiply matrices.
Finding the Determinant
Knowing how multiplication of matrices can be a pain, mathematicians instead focus their energies on operations with square matrices: matrices with the same number of rows and columns. For these matrices, the result of adding, subtracting, and multiplying will always end up with a matrix of the same size (but keep in mind that multiplication is still not always commutative).
As I have mentioned earlier, there is no division between matrices. Instead of “dividing” two matrices, we multiply the first matrix by the “inverse” of the second matrix. Finding the inverse of a matrix (which can only be done for square matrices) is beyond the scope of what we will cover today. However, not all matrices can be inverted, and deciding whether a matrix can be inverted or not is as simple as finding its determinant.
To find the determinant of a square matrix, you should, again, let the calculator do the work for you. If A is a square matrix, the determinant is normally denoted as set A or |A|. Here are videos on how to find the determinant on your TI-84 or TI-Nspire.
The determinant is just a number. If the value is 0, then the matrix cannot be inverted. If it’s not 0, then there exists an inverse, normally denoted .
Translating a Problem into a Matrix
Because a matrix is two-dimensional, each entry represents a value from two different categories. Think of it like a spreadsheet: the columns and rows of a spreadsheet normally represent two different ideas, and each entry must represent both. Here is an example:
If the spreadsheet above represents the number of items sold at a store, then we can see that the value of 5 in the middle represents “5 pieces of model Y were sold on day 2”. We can convert the information into the following matrix:
Where the columns represent the day, while the rows represent the model.
Another common way to apply matrix is to solve a system of linear equations. For example:
This system of equation can be rewritten as:
The first matrix is called the coefficient matrix, which is the most important part of the equation. The key to coming up with the coefficient matrix is to make sure that your variables in each equation are in the same order, and that the constant is on the right-hand side of the equation. The order of the equations does not matter as much, meaning you can interchange the two rows as long as you interchange the constants also.
The equations can be solved if we are able to “get rid of” the coefficient matrix. “Getting rid of” a matrix involves multiplying by its inverse. Thus, without actually solving for and , we can find out whether the system is solvable or not by finding the determinant of the coefficient matrix. If the determinant is 0, then there will be no solution as the matrix is not invertible. If the determinant is not 0, we will be able to find a unique solution.
I am intrigued… what else can I learn about matrices?
I am glad you ask that! Matrix is a very useful mathematical tool; it is studied extensively in linear algebra. Applying it with coordinate geometry and vector, we can perform some complex transformation in 2- and 3-dimensional space. It is used in engineering and computer programming (particularly in computer graphics). Come join me if you want to be The One to take down the matrix (renaming yourself as Neo will not be necessary).
As a parting gift, here is a “play on matrix”, courtesy of xkcd.com.
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