Two matrices are equal if they have the same dimensions and if the corresponding elements are equal. The second thing that's important about matrices is how to find the determinant. A determinant can only be found for a square matrix (one that has the same dimension for row width and column height e.g.: 2 X 2, 5 X 5). The determinant is a single number whose value can be found by cross multiplying, then subtracting the products:
Several operations can be performed with matrices. These include addition of two or more matrices, subtraction of two matrices, multiplication by a scalar, and multiplication of two or more matrices. It is also possible to find the inverse of a matrix. Addition and subtraction are two of the easiest operations to perform with matrices. These operations can only be performed between matrices with equal dimensions. To add two or more matrices, just add the corresponding elements. Corresponding elements are two or more numbers which are in the same row and the same column of their respective matrices (for example, two elements which would correspond are numbers located in the third row and fourth column of two different matrices). Here are some sample problems:
A matrix can be multiplied by a scalar (a number) by multiplying each element of the matrix by the scalar. In these examples, 12 and -2 are the scalars.
Multiplying matrices with each other is a bit more complicated. But first there is a simple rule we can use before each time we multiply to determine the dimensions of the product matrix. This rule will also see when multiplication is impossible. Unlike addition and subtraction, the matrices we multiply together do not need to have the same dimensions. However, the number of columns in Matrix A, for example, must be the same as the number of rows in matrix B, and so on. Multiplication of a 1 X 2 matrix and a 2 X 3 Matrix because the inner dimensions (the two twos are inner numbers when you write the dimensions of the matrices next to each other) match up. It would not be possible however to multiply a 2 X 1 matrix with a 2 X 2 matrix because the numbers of columns in the first matrix (1) is not the same as the number of rows in the second matrix (2). Not we will see why these dimensions have to match up. You begin by multiplying the elements in the first row of Matrix A by the corresponding elements in the first column of Matrix B. For example, to get the total product for row 1, Matrix A X column 1, Matrix B, you must multiply the corresponding elements then add the products of each pair of corresponding elements. This result goes in the product matrix at the row you used from Matrix A and the column you used for Matrix B. This is how the process works:
It is also possible to multiply more than two matrices together. It may look more complicated to multiply three matrices, for example, but the process is the same. Finding the inverse of a matrix will be very important when using matrices to solve "real world" problems, but it is relatively simple to do. The inverse of a matrix can be easily identified when it is multiplied by the original matrix because the product will be the identity matrix. To find the inverse of a matrix, follow this pattern:
Matrices are very useful because they are a fast and easy was of solving systems of linear equations. The system ax + by = c dx + ey = f can be written as a product of matrices. We want to solve for values of x and y, so we will first make a variables matrix, Matrix V. We will then make a coefficients matrix using the coefficients a, b, d and e. This is called the coefficients matrix, we will call it Matrix C. The one matrix left to make is the answers matrix, consisting of the answers, c and f. This will be called Matrix A.
Multiplying them together, we get CV=A. To find the solution, we will manipulate this equation to get Matrix V by itself. Since multiplying by the inverse is the same as dividing, we multiply both sides by the inverse of C and end up with V=C-1A, as seen above There's another way we can use matrices to solve systems of equations. Cramer's rule lets us use the determinant to solve for a value of x or y. The value of x is equal to a numerator matrix divided by the determinant of x. We start of by writing Matrix C, and putting its determinant it in the denominator. The matrix in the numerator is similar to the determinant of C, but slightly varied. First copy the determinant of Matrix C (don't solve this out yet!)into the numerator (so you have the same thing on top and bottom), the slide the two values from the A Matrix into the X column of that top matrix. Evaluate both numerator and denomination determinant, divide the two values, and you've got the value of x! Now just substitute that in to find y. This is what the pattern looks like:
Now let's try to solve this sample application problem using both Cramer's Rule. It is time for Sally to paint the Ferris wheel chairs at the amusement park. The park stocks two paints that can be used -- a paint with 20% oil and a higher quality paint with 35% oil. To do a quality job and also be economical, Sally plans to combine paints to get 120 gallons of paint with 30% oil. How should she combine the available paints? Let x= the number of gallons of 20% oil Let y= the number of paints of 35% oil
