Objectives
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Begin to view matrices as objects similar to numbers, which can be added, subtracted, and multiplied by a number.
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Be able to perform matrix addition and multiplication by a number and use properties of these operations.
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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.
1.
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?
2.
What do you think a square matrix is? What about a row matrix or column matrix?
3.
Let \(A=\begin{bmatrix} 1 \amp 3 \\ 2 \amp 5 \\ -1 \amp 1 \end{bmatrix}\text{,}\) \(B=\begin{bmatrix} 4 \amp -1 \\ -2 \amp -3 \\ 2 \amp 0 \end{bmatrix}\text{,}\) and \(C=\begin{bmatrix} 2 \amp 0 \\ -1 \amp 5 \\ -2 \amp 0 \end{bmatrix}\text{.}\)
Calculate each of the items below. What do you notice about your answers?
(a)
(b)
(c)
\(5(A+B)\)
(d)
4.
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.
Let \(A\text{,}\) \(B\text{,}\) and \(X\) be matrices of the same size, and let \(k\) and \(n\) be numbers.
(a)
\(A-B=A+(-B)\)
(b)
\((k+n)A = kA+nA\)
(c)
(d)
(e)
(f)
\(kB=Bk\)
(g)
\(AB=BA\)
(h)
For each matrix \(C\text{,}\) there is a matrix \(Z\) such that \(Z+C=C+Z=C\text{.}\) (The matrix \(Z\) acts like the number \(0\) does for numbers.)
(i)
For each matrix \(C\text{,}\) there is a matrix \(I\) such that \(IC=CI=C\text{.}\) (The matrix \(I\) acts like the number \(1\) does for numbers.)
