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A matrix is a rectangular array of numbers that are enclosed by brackets. When working with matrices (the plural form of a matrix), it''s important to understand their dimensions, particularly if you want to perform operations like addition and subtraction. Let''s go over the basics of matrix dimensions.
Each matrix is made up of one or more numbers (also called entries or elements) that line up in rows (horizontal) and columns (vertical). Here is an example of a matrix:
$\left[\begin{array}{cc}3& 5\\ 99& -0.5\end{array}\right]$The dimensions of a matrix are the number of rows by the number of columns. When writing matrix dimensions, it''s important to always write the number of rows first and the number of columns second. If a matrix has a rows and b columns, its dimensions would be $a\times b$ -- or it would be considered an $a\times b$ matrix.
In the example above, the matrix is a $2\times 2$ matrix because it has 2 rows and 2 columns. Since it also has an equal number of rows and columns, it would be called a square matrix.
Now, what would you say are the dimensions of the following matrix?
$\left[\begin{array}{cc}5& 8\\ 4& 3\\ 9& -2\end{array}\right]$Since this matrix has 3 rows and 2 columns, its dimensions are $3\times 2$ .
Let''s try another one. What are the dimensions of this matrix?
$\left[\begin{array}{cccc}\frac{1}{3}& 7& \frac{2}{3}& 6\end{array}\right]$This matrix has 1 row and 4 columns, making it a $1\times 4$ matrix. It is also called a row matrix (a matrix with one row and any number of columns).
What are the dimensions of this matrix?
$\left[\begin{array}{c}-5\\ 3\\ 2\\ 7\\ -4\end{array}\right]$This matrix, also known as a column matrix (a matrix with one column and any number of rows), has 5 rows and 1 column, which makes it a $5\times 1$ matrix.
A matrix can only be added to or subtracted from when it has the same dimensions as another matrix. For example, you can''t add a $2 \times 3$ matrix to a $3\times 3$ matrix. However, you can add two $2\times 3$ matrices or two $3\times 3$ matrices.
Let''s look at some examples.
Adding matrices
Adding matrices requires that you add corresponding entries in each matrix, and then place the sum in the corresponding position in the resulting matrix.
Let''s add these two matrices:
$\left[\begin{array}{cc}3& -2\\ 1& 4\end{array}\right]+\left[\begin{array}{cc}-1& 8\\ 4& 3\end{array}\right]$Before getting started, we want to make sure the matrices have the same dimensions. Since they''re both $2\times 2$ matrices, we''re all set.
$\left[\begin{array}{cc}3& -2\\ 1& 4\end{array}\right]+\left[\begin{array}{cc}-1& 8\\ 4& 3\end{array}\right]=\left[\begin{array}{cc}3+\left(-1\right)& -2+8\\ 1+4& 4+3\end{array}\right]$Than resulting matrix is:
$\left[\begin{array}{cc}2& 6\\ 5& 7\end{array}\right]$Subtracting matrices
In order to subtract matrices, you''ll subtract the entries in the second matrix from the corresponding entries in the first matrix. Then, you''ll place the difference in the corresponding position in the resulting matrix.
Let''s try an example:
$\left[\begin{array}{cc}9& 5\\ 1& 8\\ -5& 9\end{array}\right]-\left[\begin{array}{cc}1& 6\\ 6& 7\\ -3& 2\end{array}\right]$Again, we want to start by making sure that the two matrices have the same dimensions. Since they''re both $3\times 2$ matrices, we''re all set to begin.
$\left[\begin{array}{cc}9& 5\\ 1& 8\\ -5& 9\end{array}\right]-\left[\begin{array}{cc}1& 6\\ 6& 7\\ -3& 2\end{array}\right]=\left[\begin{array}{cc}9-1& 9-1\\ 1-6& 8-7\\ -5-\left(-3\right)& 9-2\end{array}\right]$Than resulting matrix is:
$\left[\begin{array}{cc}8& -1\\ -5& 1\\ -2& 7\end{array}\right]$Note: When multiplying matrices, their dimensions must be compatible, which means, the number of columns in the first matrix must have the same as the number of rows in the second matrix. For example, $\left(2\times 5\right)\times \left(5\times 3\right)$ would be compatible, while $\left(5\times 3\right)\times \left(2\times 5\right)$ would not.
$\left[\begin{array}{ccc}5& 4& 8\\ 4& -7& 3\\ 9& 2& -2\end{array}\right]$
Answer: $3\times 3$
$\left[\begin{array}{cccccc}0.3& -8& 27& 7& 2& -5\\ 57& 1& 14& -25& 4& 80\end{array}\right]$
Answer: $2\times 6$
$\left[\begin{array}{c}3\\ 1\\ 9\\ 6\\ 7\end{array}\right]$
Answer: $5\times 1$
$\left[\begin{array}{ccccc}6& -7& 0.4& 8& -3\end{array}\right]$
Answer: Row matrix
$\left[\begin{array}{cc}1& 6\\ 3& 5\end{array}\right]+\left[\begin{array}{cc}1& 3\\ -6& 4\end{array}\right]=\left[\begin{array}{cc}2& 9\\ -3& 9\end{array}\right]$
Answer: $\left[\begin{array}{cc}2& 9\\ -3& 9\end{array}\right]$
$\left[\begin{array}{cc}-4& 9\\ 7& -1\end{array}\right]+\left[\begin{array}{cc}-5& 2\\ -2& 6\end{array}\right]=\left[\begin{array}{cc}1& 7\\ 9& -7\end{array}\right]$
Answer: $\left[\begin{array}{cc}1& 7\\ 9& -7\end{array}\right]$
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