When we were children, we learned about numbers and their names and numerals, then how to count, how to add, subtract, and multiply, and various properties of those operations that led us to be able to solve equations containing unknown numbers. All of this took years to became second nature, but we will build on the knowledge and familiarity with how numbers work to understand how matrices work.
We defined a matrix in Definition 1.2.6, and included defining rows, columns, and notation for the size of a matrix and for the entries within a matrix. An augmented matrix, whose entries represent the coefficients and constants in a linear system, is a type of matrix or a way of interpreting a matrix, but when we look at a matrix we don’t have to think of a linear system at all. A matrix is an array of numbers; both the size of the array and the contents are important to the essence of a matrix.
Because numbers can be added in either order and the answer is the same, for example \(2+3=3+2=5\text{,}\) it follows that adding matrices in either order results in the same matrix also.
Subtraction and multiplying by numbers can be related back to addition, for example \(3-1=3+(-1)\) and \(3\cdot 5 = 5+5+5\text{,}\) so it makes sense that subtracting two matrices and multiplying a matrix by a number also work by doing the operation to each entry in the matrix.
We know that numbers can sometimes appear in different forms but really be the same number: for example, \(2\text{,}\)\(\frac{10}{5}\text{,}\)\(2.00\) are all different ways to write the number “two”. Two matrices are equal if each corresponding position contains the same number, regardless of the form of the number.
Matrix \(A\) and matrix \(C\) have different sizes, so there are positions without corresponding entries to add. For example, the third column of \(A\) has nothing corresponding in \(C\text{.}\)
We never had to worry about whether two numbers could be added or subtracted, so that is a difference between numbers and matrices. The size of a matrix is key to being able to perform calculations; only matrices of the same size can be added and subtracted, and if two matrices have different sizes, then they cannot be equal.
When the dimensions are clear from the context, we just write \(\mathbf{0}\) in bold font and call it the zero matrix, even though there is a different zero matrix for each size.
Note the difference between \(0\) and \(\mathbf{0}\text{:}\) the first is a number and the second is a matrix in which every entry is the number \(0\text{.}\)
You might have noticed that there are a couple of operations we do with numbers that we have not discussed how to do with matrices, namely, multiplication and division. There’s a very good reason for this: multiplication and division in the realm of matrices are very different from multiplication and division of numbers, and we leave them for future sections.
In your own words, explain how to add two matrices together. Be sure to discuss if addition is always possible or if not, what conditions need to be satisfied.
There is one matrix of all zeros for each size. However, \(\begin{bmatrix} 0 \amp 0 \\ 0 \amp 0 \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}\text{,}\) for example.
False.
There is one matrix of all zeros for each size. However, \(\begin{bmatrix} 0 \amp 0 \\ 0 \amp 0 \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}\text{,}\) for example.
We have discussed multiplying a matrix by a number. Most texts will use the term scalar instead of “number”. There are good reasons for this, but they are outside the scope of this material. Still, it is good to recognize both words could be used.
Consider the matrices \(A=\begin{bmatrix}1 \amp 2 \amp 3 \end{bmatrix}\) and \(B=\begin{bmatrix} 1\\ 2 \\ 3 \end{bmatrix}\text{.}\) Say why \(A\) and \(B\) are not equal as matrix objects. Does this make sense to you, or do you think they should be the same thing?
Consider the list of proposed properties below. For each one, say whether it is True, False, or Insufficient Information. Note that these are all statements you might be used to thinking about with numbers, and the goal is to think about how matrices and numbers are similar in some ways and different in some ways.
Many of the properties we’re used to still hold for matrix addition and multiplication by a number, such as associativity and commutativity of addition and the distributive property.
During the month of January, “ABC Appliances” sold \(33\) microwaves, \(17\) refrigerators, and \(33\) stoves, while “XYZ Appliances” sold \(57\) microwaves, \(27\) refrigerators and \(31\) stoves.
During the month of February, “ABC Appliances” sold \(38\) microwaves, \(27\) refrigerators, and \(40\) stoves, while “XYZ Appliances” sold \(51\) microwaves, \(20\) refrigerators and \(34\) stoves.
Exercise2.1.4.Adding Matrices, List of Lists Notation.
Many programming languages, including Python and Sage, use a list of lists to enter matrices. We use that notation in this problem instead of the usual array answer box notation to avoid giving away information about the size of the matrix or whether such a matrix exists at all.
