Matrices are sets of numbers set in brackets, and they can be added, subtracted, and multiplied. they can help you solve systems of equations and each matrix has its own determinant. The numbers in a matrix are called elements
The Zero Matrix: These are all zero matrices.All the elements are zeroes. Pretty self-explanatory.
The Identity Matrix: This matrix has 1's on the main diagonal and has zeroes everywhere else. It works like how the number 1 works in multiplication- multiply it with any matrix and the product will be the same matrix. It will always be a square matrix.
Adding and Subtracting Matrices: To add matrices, simply add the numbers in each matrix that have the same row and column numbers. For subtracting, do the same but subtract instead of add. Two matrices must be the same size with the same number of rows and columns in order to add and subtract.
Problem Examples: Add the following matrices.
2x3 matrix with elements 3,4,7,7,8,8
2x2 matrix with elements 11,8,-4,2
2x2 matrix with elements 3,4,0,2
Scalar Multiplication: Lets say you have a 2x2 matrix [A]. When looking at 2[A], the 2 is called a scalar. If each of the elements in [A] were ones, all the elements in 2[A] would be twos. If [A] had the elements 1,3,2,5, the elements of 2[A] would be 2,6,4,10. In other words, each element of the matrix is multiplied by the scalar.
Multiplying Matrices: In order to multiply matrices, the number of columns of the first matrix must equal the number of rows in the second matrix. So a 2x4 matric could be multiplied by a 4x3 matrix, but not to a 3x2 or 3x4 matrix. After making sure the matrices you are multiplying follow this rule, you multiply the elements of each row of the first matrix by the elements of each column in the second matrix. you then add these products. Note that AxB does not necessarily equal BxA in matrices.
Determinants of Matrices: The determinant of a matrix is a real number corresponding to a particular matrix that can be found using a number of methods. The determinant can only be found for square matrices, like 2x2 or 4x4. The method for 2x2 matrices is a bit different for anything with larger dimensions than that. If you have a 2x2 matrix a, b, c, d, the determinant for that matrix is ad-bc. You are being asked to find a determinant if the matrix is no longer in brackets but has straight lines on either side, like |A| instead of [A].
Problem Examples: Find the determinants of the following matrices
![Picture](/uploads/4/3/9/7/43974927/5321874.gif?96)
(1 x 4)-(2 x 3)=-2
(1 x 4)-(2 x 3)=-2
![Picture](/uploads/4/3/9/7/43974927/1661568.gif?107)
(2 x 3)-(1 x -1)=7
Expansion by minors: To find the determinant of a 3x3 matrix, or a 4x4 matrix, or anything higher, a different approach must be taken. One of the ways for a 3x3 matrix can be seen below. This method is easy if you know how to find the determinant of a 2x2 matrix. If you like finding determinants of smaller matrices, you may prefer this method. You can choose any row or column you like, and go element by element crossing out the perpendicular line that that element is in. For each one, record the element and the smaller matrix created by your lines. You then add or subtract the values you get, but remember the signs need to alternate- if you begin with adding, you must then subtract, then add again. You are adding or subtracting the determinant of the smaller matrices you have created multiplied by the element you used to create it.
This next way of finding determinants can be used for 3x3, 4x4, etc., and it is essentially the same as a 2x2, just drawn out a bit more and is a bit more complicated.
Solving Systems of Equations: You can use matrices to solve systems of equations. One way to do this is to make your system into matrices- the coefficients on one side, all in the right order and the same variables going down, and the answers on the right side. You can then use a series of rules to move and manipulate the numbers until on the left side to become the identity matrix, and the numbers left on the right are your solution. This can be complicated without a calculator, and there are easier ways to solve the systems if you are doing it by hand. You can also use the inverse method, which also includes forming your system into matrices.
Cramer's Rule: Cramer's rule can also help to solve systems of equations, but this method uses determinants. Again, you want to set up your matrices using your systems, but you will substitute certain columns for certain reasons. First, simply find your determinant D for the coefficient part of your system. For the problem 2x-3y=3 and 4x-2y=10, your determinant D is 2, -3, bottom row 4, -2. When you solve, you get 8. Now to find your x-value, look at the same determinant, but substitute the x column (the first one) for the solution values. Now instead of 2, -3,4, -2, you have 3, -3, bottom row 10, -2. The x coefficients 2 and 4 have been replaced with the answer column, 3 and 10. To find the x-value for this system, you find the solution for this new determinant, Dx, and divide it by your determinant D. You can do the same for y, just replace the y column with the original answers (3 and 10). This can be used for systems with more than one variable as well.
(Mr. Green you said not to worry about a common core problem for matrices)