Conceptual Explanations: Matrices

A “matrix” is a grid, or table, of numbers. For instance, the following matrix represents the prices at the store “Nuthin’ But Bed Stuff.”

(The matrix is the numbers, not the words that label them.)

Of course, these prices could be displayed in a simple list: “King-sized mattress,” “Queen-sized mattress,” and so on. However, this two-dimensional display makes it much easier to compare the prices of mattresses to box springs, or the prices of king-sized items to queen-sized items, for instance.

Each horizontal list of numbers is referred to as a row; each vertical list is a column. Hence, the list of all mattresses is a row; the list of all king-sized prices is a column. (It’s easy to remember which is which if you think of Greek columns, which are big posts that hold up buildings and are very tall and...well, you know...vertical.) This particular matrix has 5 rows and 3 columns. It is therefore referred to as a 5×3 (read, “5 by 3”) matrix.

If a matrix has the same number of columns as rows, it is referred to as a square matrix.

Adding and Subtracting Matrices

Adding matrices is very simple. You just add each number in the first matrix, to the corresponding number in the second matrix.

123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {}++ 605040302010605040302010 size 12{ left [ matrix { "60" {} # "50" {} # "40" {} ## "30" {} # "20" {} # "10"{} } right ]} {}== 615243342516615243342516 size 12{ left [ matrix { "61" {} # "52" {} # "43" {} ## "34" {} # "25" {} # "16"{} } right ]} {}

For instance, for the upper-right-hand corner, the calculation was 3 + 40 = 43 3+40=43. Note that both matrices being added are 2×3, and the resulting matrix is also 2×3. You cannot add two matrices that have different dimensions.

As you might guess, subtracting works much the same way, except that you subtract instead of adding.

605040302010605040302010 size 12{ left [ matrix { "60" {} # "50" {} # "40" {} ## "30" {} # "20" {} # "10"{} } right ]} {} –– 123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {} == 5948372615459483726154 size 12{ left [ matrix { "59" {} # "48" {} # "37" {} ## "26" {} # "15" {} # 4{} } right ]} {}

Once again, note that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have different dimensions.

Multiplying a Matrix by a Constant

What does it mean to multiply a number by 3? It means you add the number to itself 3 times.

Multiplying a matrix by 3 means the same thing...you add the matrix to itself 3 times.

33 123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {} == 123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {} ++ 123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {} ++ 123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {} == 369121518369121518 size 12{ left [ matrix { 3 {} # 6 {} # 9 {} ## "12" {} # "15" {} # "18"{} } right ]} {}

Note what has happened: each element in the original matrix has been multiplied by 3. Hence, we arrive at the method for multiplying a matrix by a constant: you multiply each element by that constant. The resulting matrix has the same dimensions as the original.

1212 123456123456 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6{} } right ]} {} == 121322523121322523 size 12{ left [ matrix { { {1} over {2} } {} # 1 {} # { {3} over {2} } {} ## 2 {} # { {5} over {2} } {} # 3{} } right ]} {}

Matrix Equality

For two matrices to be “equal” they must be exactly the same. That is, they must have the same dimensions, and each element in the first matrix must be equal to the corresponding element in the second matrix.

For instance, consider the following matrix equation.

1x+y12101x+y1210 size 12{ left [ matrix { 1 {} # x+y {} ## "12" {} # "10"{} } right ]} {} == 118x−y10118x−y10 size 12{ left [ matrix { 1 {} # "18" {} ## x - y {} # "10"{} } right ]} {}

Both matrices have the same dimensions. And the upper-left and lower-right elements are definitely the same.

But for the matrix to be equal, we also need the other two elements to be the same. So

x + y = 18 x+y=18

x – y = 12 x–y=12

Solving these two equations (for instance, by elimination) we find that x = 15 x=15, y = 3 y=3.

You may notice an analogy here to complex numbers. When we assert that two complex numbers equal each other, we are actually making two statements: the real parts are equal, and the imaginary parts are equal. In such a case, we can use one equation to solve for two unknowns. A very similar situation exists with matrices, except that one equation actually represents many more statements. For 2×2 matrices, setting them equal makes four separate statements; for 2×3 matrices, six separate statements; and so on.

OK, take a deep breath. Even if you’ve never seen a matrix before, the concept is not too difficult, and everything we’ve seen so far should be pretty simple, if not downright obvious.

Let that breath out now. This is where it starts to get weird.