Matrices and Determinants

Today we are going to talk about “Matrices and Determinants”. I will explain you two different ways of solving an equation system by using matrices and how to do simple calculations like multiplication and addition with matrices.

If we have an equation system like this:

1x+1y-1z= 6

3x-1y+3z=11

4x+2y-3z=14

we can solve it by using the two methods we learned to solve linear equation systems with two unknown variables but this would be very difficult. An idea to simplify the system would be to write only the numbers and to leave out the variables. We get a matrix with 3 rows and 4 columns:

1 1 -1 6

2 -1 3 11

4 2 -3 14

It is a rule, that you can add any row multiplied with any number to any other row without changing the matrix’ worth, so we add -2 multiplied with the middle row to the last row. The aim of this operation is, to get a zero in the column representing the x.

1 1 -1 6

2 -1 3 11

0 4 -9 -8

Our next operation is, to add -2 multiplied by the 1^st row to the middle row, leaving the middle row with a zero in the first column.

1 1 -1 6

0 -3 5 -1

0 4 -9 -8

Finally, we add 4/3 multiplied by the middle row to the bottom row and so we have finished the so-called triangulariztion process. We get the matrix

1 1 -1 6

0 -3 5 -1

0 0 -2^1/3 -9^1/3

Now we can fill in the x’s y’s and z’s again, and we see that the equation system is nearly solved:

1x +1y -1z =6

-3y +5z =-1

-2^1/3z =-9^1/3

Now we can easily solve the equation system..................we get the result: z=4; y=7; x=3

Matrices are classified according to the number of rows and columns they contain. A matrix like this: [2x2 Matrix] is called a 2 by 2 matrix. This one is called a 2 by 3 matrix. It has 2 rows and 3 columns. A matrix that has as many rows as columns is called a square matrix.

We use capital letters (A,B) to represent a matrix in the same way we used small letters to represent a number or a vector. A matrix consisting of all zero’s is a very special one. It is called a zero matrix.

When I want to add the matrix A=
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and the matrix B=
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I only have to add the corresponding elements to get: A+B=
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=
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The matrix multiplication is a little bit more complicated. If we have the matrix A=
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and the matrix B=
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we multiply them by multiply each element of a row of the first matrix with the corresponding element of a row of the 2^nd matrix and add them.

For example A*B=
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=
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Matrices can only be multiplied when the have the same number of rows! The matrix I=
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is called the identity matrix. Concerning multiplication, it corresponds at 1 in ordinary numbers, so that I x A = A

In fifth class, we learned about Cramer’s rule, a procedure to find the solutions of any kind of equation system. We will do the following example:
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written as a system of three matrices we have
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Now we have to calculate the determinant of the 3 by 3 matrix. For a two by two matrix the procedure is very simple. A=
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We only have to calculate ad-bc=D

for calculating the 3 by three matrix we can use two different ways.

The first is, to write:
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The determinant is:
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The second method you can use for any kind of square matrix. We start with the first element in the first row and multiply it with the determinant of its minor. Then we subtract the product of the second element and the determinant of its minor. Then add the product of the third element and its minor.

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This method will work for any kind of square matrix. If you want to calculate a 4 x 4 matrix you have to calculate four 3 x 3 matrices before. As you see, the calculation of determinants is very complicated and a lot of work, that’s why it is useful to use computers for this mathematical operation.

But now back to our equation system.
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To find the solution of x, we replace the coefficients of x in the coefficient matrix with the column of constants from the right hand side. Then we use the determinant as numerator of the solutions for x. We follow the same procedure to find the solutions for y and z:

ΠWrite down the matrix with 3 rows and 4 columns to represent the equation. Each row represents one equation. The first column contains the coefficients of x; the second column contains the coefficients of y and the third column contains the coefficients of z. The 4^th column contains the terms from the right hand side of the system.

Next, triangularize the matrix. The basic move is to multiply one row by a number and then add that row to another row. Once the triangularization has been completed, solve for z first, then y, and then x.

2 by 2 matrix (2 x 2):
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It has two rows and two columns

2 by 3 matrix (2 x 3):
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It has 2 rows and 3 columns

Square matrix (n x n):
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It has as many rows as columns

Matrix addition (subtraction):
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=
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Matrix multiplication:

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=
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A=
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; D(A)= ad-bc

A=
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;
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Used literature:

Reichel - Müller - Laub; “Lehrbuch der Mathematik 5”

Reichel - Müller - Laub; “Lehrbuch der Mathematik 6”

....................................; “Matrices and Determinants”