Dealing with matrix operations can be daunting at first because of the common feeling that you must keep track of a large amount of numbers. Some students attempt to add and multiply matrices by brute force, keeping all the numbers in their heads. However, simplifying the processes can not only make matrix operations easier, but also make you more accurate in computing them.
Multiply scalars -- the lone numbers in front of matrices -- first. Look for numbers on their own, not in matrices themselves, sitting next to matrices. A scalar is a just a number, such as those you are used to dealing with in lower-level math. When you see the expression 2x3, you are multiplying two scalars to get a new scalar 6. In matrix algebra, a scalar works the same way but multiplies an entire matrix -- that is, every element inside the matrix. For example, if B represents a matrix, 2B is a scalar times a matrix. In this case, you would multiply every element in B by the number 2, giving you a new matrix. For example, if the first row of matrix B is [3, 4], the new row will be [6, 8].
Rewrite the matrix problem with scalar-multiplied matrices. Replace the old matrix with the new one in the problem. For example, if your problem is AB + 2B, where A and B are matrices, do 2B first and replace it with the new matrix, in which all elements are doubled. The problem now becomes AB + C, where C is the new matrix.
Perform multiplication by “lining up” rows and columns. Multiply AB by taking the first row of A “lining it up” with the first column of B. Multiple across the lines and add. This gives you the first element of the new matrix. For example, if the first row of A is [5, 0] and the first column of B is [4, 1], lining up the row and column will put 5 and 4 next to each other and 0 and 1 next to each other. The multiplication then becomes more obvious: 5_4 = 20 and 0_1 = 0. Adding these together gives 20, the first element of the new matrix.
Rewrite the matrix problem with multiplied matrices. In the problem AB + C, rewrite AB as D, which is the matrix you get after multiplying A and B.
Add or subtract matrices by putting all the numbers of individual matrices into equations within one big matrix. Rewrite the problem, such as A + B as a single matrix that takes the elements from A and the elements from B, placing them in a big matrix. Use plus signs to separate the numbers for addition and minus signs for subtraction. For example, if the first row of A is [2, 1] and the first row of B is [10, 4], place these numbers in the first row of the new, big matrix as [2+10, 1+4]. Perform the addition after you have rewritten the matrix. This can help you avoid making small mistakes when adding or subtraction in your head.
Technically, a scalar is a matrix with a single element, which is why it has a special name -- scalar -- in spite of it being so familiar to students as "just a number." But when you hear the word "scalar" in matrix algebra, you can just think "number," if it helps.