Applicable Mathematics/Matrices

Matrices

A matrix is a rectangular array of numbers enclosed in brackets. In a notational sense, what differentiates a list of numbers from a matrix is its format. The numbers are listed so that each number has a certain, specific position between the brackets. Each number, or value, in a matrix is called an entry.

One of the main benefits of matrices is the properties which allow them to be manipulated and used for many different, but useful purposes.

Matrices can vary in size. This variation in size is called dimensions. Just like the dimensions of a room (width x length) matrices have dimensions (# of rows x # of columns). Thus, a 2 x 3 (read 2 by 3) matrix will have 2 rows and 3 columns.

Example of a 2 x 3 matrix:

$M=\left[\begin{array}{ccc}572& 98& 302\\ 1& 732& 22\end{array}\right]$

Another term associated with matrices is address. Like your home address, an address describes where each value, or entry, of a matrix lives. The address is composed of the lowercase letter of the matrix with the row and column number (in that order) as subscripts.

Using the 2 x 3 matrix M as an example, the positions of the values are as follows:

$M=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\end{array}\right]=\left[\begin{array}{cccc}572& 98& 302& \\ 1& 732& 22\end{array}\right]$

$a_{11}=572a_{12}=98a_{13}=302a_{21}=1a_{22}=732a_{23}=22$

A square matrix is any matrix that has the same number of rows as it does columns.

Example: 2 x 2 or 3 x 3 matrices are both square matrices.

$2x2=\left[\begin{array}{cc}1& 12\\ 21& 2\end{array}\right]3x3=\left[\begin{array}{ccc}7& 9& 30\\ 1& 2& 2\\ 5& 43& 6\end{array}\right]$

Notice the numbers in red above in the 2 x 2 and 3 x 3 square matrices. These numbers are in the addresses of the main diagonal. The main diagonal of a square matrix is the diagonal from the upper left corner entry to the bottom right corner entry. Notice that only square matrices can have a main diagonal.

Adding and Subtracting Matrices

To add or subtract matrices, the sum or difference is found when addition or subtraction is applied to corresponding entries.

For example,

$\left[\begin{array}{cc}7& 4\end{array}\right]+\left[\begin{array}{cc}3& 9\end{array}\right]=\left[\begin{array}{cc}10& 13\end{array}\right]$

$\left[\begin{array}{cc}{a}_{11}& a_{12}\end{array}\right]+\left[\begin{array}{cc}{b}_{11}& b_{12}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}+b_{11}& a_{12}+b_{12}\end{array}\right]$

$\left[\begin{array}{cc}7& 4\end{array}\right]-\left[\begin{array}{cc}3& 9\end{array}\right]=\left[\begin{array}{cc}4& -5\end{array}\right]$

$\left[\begin{array}{cc}{a}_{11}& a_{12}\end{array}\right]-\left[\begin{array}{cc}{b}_{11}& b_{12}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}-b_{11}& a_{12}-b_{12}\end{array}\right]$

Since addition or subtraction takes place using corresponding entries, matrices must have the same dimensions in order to complete either operation.

Consider this operation

$\left[\begin{array}{c}52\\ 36\end{array}\right]+\left[\begin{array}{ccc}12& 16& 5\end{array}\right]====>\left[\begin{array}{c}{a}_{11}\\ a_{21}\end{array}\right]+\left[\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\end{array}\right]CA{N}^{\prime }TBEDONE$

Now consider these matrices

$\left[\begin{array}{cc}52& 4\\ 36& 18\end{array}\right]+\left[\begin{array}{cc}12& 16\\ 34& 2\end{array}\right]====>\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]+\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}+b_{11}& a_{12}+b_{12}\\ a_{21}+b_{21}& a_{22}+b_{22}\end{array}\right]=\left[\begin{array}{cc}64& 20\\ 70& 20\end{array}\right]$

Properties of Equality for Matrices

Commutative Property

Matrix addition is commutative $A+B=B+A$

$\left[\begin{array}{ccc}5& 9& 0\\ 1& 3& 2\end{array}\right]+\left[\begin{array}{ccc}7& 8& 3\\ 4& 7& 6\end{array}\right]=\left[\begin{array}{ccc}7& 8& 3\\ 4& 7& 6\end{array}\right]+\left[\begin{array}{ccc}5& 9& 0\\ 1& 3& 2\end{array}\right]$

Associative Property

Matrix addition is associative. $A+B+C=(A+B)+C=A+(B+C)$

$\left[\begin{array}{cc}4& 7\\ 1& 8\end{array}\right]+\left[\begin{array}{cc}3& 18\\ 24& 7\end{array}\right]+\left[\begin{array}{cc}1& 8\\ 14& 6\end{array}\right]=(\left[\begin{array}{cc}4& 7\\ 1& 8\end{array}\right]+\left[\begin{array}{cc}3& 18\\ 24& 7\end{array}\right])+\left[\begin{array}{cc}1& 8\\ 14& 6\end{array}\right]=\left[\begin{array}{cc}4& 7\\ 1& 8\end{array}\right]+(\left[\begin{array}{cc}3& 18\\ 24& 7\end{array}\right]+\left[\begin{array}{cc}1& 8\\ 14& 6\end{array}\right])$

Additive Identity

The zero matrix is the additive identity matrix 'O'.

A Zero matrix is a matrix in which all of the entries are zero. $O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Example of additive identity $A+O=A$

$\left[\begin{array}{cc}7& 4\\ 6& 9\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}7& 4\\ 6& 9\end{array}\right]$

Additive Inverse

It takes two matrices to form a pair of inverses. Two matrices are additive inverses if their sum is the zero matrix. This occurs when the additive inverse of a matrix contains the values opposite of each entry. $A+(-A)=O$

$\left[\begin{array}{ccc}8& 2& -1\\ -6& 9& -3\end{array}\right]+\left[\begin{array}{ccc}-8& -2& 1\\ 6& -9& 3\end{array}\right]=\left[\begin{array}{ccc}8+-8& 2+-2& -1+1\\ -6+6& 9+-9& -3+3\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$

Multiplicative Identity Matrix

A multiplicative identity matrix, usually denoted by the letter I, is any square matrix that has a value of 1 in all the entries along the main diagonal and 0 in the remaining entries.

$I_{2x2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]I_{3x3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Any matrix multiplied by an identity matrix will retain it's original entries. $A+I=A$

$A=\left[\begin{array}{ccc}6& 15& 9\\ 23& 5& 43\end{array}\right]I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ ${\displaystyle }$ $AI=\left[\begin{array}{ccc}(6)(1)+(15)(0)+(9)(0)& (6)(0)+(15)(1)+(9)(0)& (6)(0)+(15)(0)+(9)(1)\\ (23)(1)+(5)(0)+(43)(0)& (23)(0)+(5)(1)+(43)(0)& (23)(0)+(5)(0)+(43)(1)\end{array}\right]$

$AI=\left[\begin{array}{ccc}6& 15& 9\\ 23& 5& 43\end{array}\right]$

Multiplicative Inverse Matrix

If matrices $AxB=I$ where I is the identity matrix, then A and B are multiplicative inverses of one another.

$A=\left[\begin{array}{ccc}-1& 0& 2\\ 4& 1& -1\\ 2& 0& 1\end{array}\right]B=\left[\begin{array}{ccc}-0.2& 0& 0.4\\ 1.2& 1& -1.4\\ 0.4& 0& 0.2\end{array}\right]$

$AB=\left[\begin{array}{ccc}(-1)(-0.2)+(0)(1.2)+(2)(0.4)& (-1)(0)+(0)(1)+(2)(0)& (-1)(0.4)+(0)(-1.4)+(2)(0.2)\\ (4)(-0.2)+(1)(1.2)+(-1)(0.4)& (4)(0)+(1)(1)+(-1)(0)& (4)(0.4)+(1)(-1.4)+(-1)(0.2)\\ (2)(-0.2)+(0)(1.2)+(1)(0.4)& (2)(0)+(0)(1)+(1)(0)& (2)(0.4)+(0)(-1.4)+(1)(0.2)\end{array}\right]$ $AB=I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Thus, matrices A and B are multiplicative inverses of each other.

Multiplying Matrices

Another useful property of matrices is called a scalar. A scalar is a number located outside of a single matrix. To apply the scalar to the matrix simply multiply each entry of the matrix by the scalar.

For example,

$3\left[\begin{array}{ccc}3& 9& 15\\ 1& 7.5& 2\end{array}\right]=\left[\begin{array}{ccc}(3)3& (3)9& (3)15\\ (3)1& (3)7.5& (3)2\end{array}\right]=\left[\begin{array}{ccc}9& 27& 45\\ 3& 22.5& 6\end{array}\right]$

In order to multiply two matrices together it is necessary to pay attention to their dimensions. Matrices A and J can be multiplied only if the number of columns in A equals the number of rows in J. Also, another hint is that the product of an a x j and a j x b matrix is an a x b matrix. Notice that the number of columns from the first matrix equals the number of rows from the second matrix (j = j).

First, we'll look at how to test to see if we can multiply matrices.

Consider matrices Q and R.

$Q=2x3R=3x23columns=3rows$

Since the number of columns of matrix Q equals the rows of matrix R, these matrices can be multiplied. This will produce a 2 x 2 matrix.

$Q=\left[\begin{array}{ccc}1& 7& 9\\ 3& 5& 1\end{array}\right]xR=\left[\begin{array}{cc}8& 9\\ 3& 5\\ 4& 6\end{array}\right]$

Multiplying Q and R.

During this process, you may find your fingers and mental addition extremely helpful. You can use your left pointer finger to follow the entries in the rows of matrix Q and your right pointer finger to follow the columns of matrix R. To multiply matrices, add the products of consecutive entries in corresponding rows of matrix Q and columns of matrix R.

$QR=\left[\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\end{array}\right]\left[\begin{array}{ccc}{r}_{11}& r_{12}& \\ r_{21}& r_{22}\\ r_{31}& r_{32}\end{array}\right]=$ $\left[\begin{array}{ccccc}Q{R}_{11}=q_{11}\bullet r_{11}+& q_{12}\bullet r_{21}+& q_{13}\bullet r_{31}Q{R}_{12}=q_{11}\bullet r_{12}+& q_{12}\bullet r_{22}+& q_{13}\bullet r_{32}\\ Q{R}_{21}=q_{21}\bullet r_{11}+& q_{22}\bullet r_{21}+& q_{23}\bullet r_{31}Q{R}_{22}=q_{21}\bullet r_{12}+& q_{22}\bullet r_{22}+& q_{23}\bullet r_{32}\end{array}\right]$

$QR=\left[\begin{array}{ccc}1& 7& 9\\ 3& 5& 1\end{array}\right]\left[\begin{array}{cc}8& 9\\ 3& 5\\ 4& 6\end{array}\right]=\left[\begin{array}{cc}(1)(8)+(7)(3)+(9)(4)& (1)(9)+(7)(5)+(9)(6)\\ (3)(8)+(5)(3)+(1)(4)& (3)(9)+(5)(5)+(1)(6)\end{array}\right]=\left[\begin{array}{cc}65& 98\\ 43& 58\end{array}\right]$

Determinants

Every square matrix has a value called a determinant. The determinant of a square matrix is the difference of the products of the diagonals.

The determinant of a 2 x 2 matrix can be found as follows:

$det\left[\begin{array}{cc}9& 2\\ 7& 8\end{array}\right]=(9)(8)-(7)(2)=58$

The "down" diagonal is in red and the "up" diagonal is in blue. The up diagonals are always subtracted from the down diagonals. Matrices that are larger than a 2 x 2 matrix become a little more complicated when finding the determinant but the same rules apply.

Let's find $det\left[\begin{array}{ccc}1& 6& 19\\ 8& 17& 5\\ 14& 9& 3\end{array}\right]$

When finding the determinant of a 3 x 3 matrix it is helpful to write the first two columns to the right side of the matrix like so,

$det\left[\begin{array}{ccc}1& 6& 19\\ 8& 17& 5\\ 14& 9& 3\end{array}\right]\begin{array}{cc}1& 6\\ 8& 17\\ 14& 9\end{array}$

As shown above in the 2x2 matrix, the numbers are color coded. The blue numbers, once again, indicate they are used in the up diagonals, the red are used in the down diagonals, and those in magenta are used in both.

$det\left[\begin{array}{ccc}1& 6& 19\\ 8& 17& 5\\ 14& 9& 3\end{array}\right]\begin{array}{cc}1& 6\\ 8& 17\\ 14& 9\end{array}=(1*17*3)+(6*5*14)+(19*8*9)-[(14*17*19)+(9*5*1)+(3*8*6)]$

$=(51+420+1,368)-(4,522+45+144)=-2872$

Thus, the determinant of the above 3x3 matrix is -2872.