There are three operations on matrices: addition, subtraction and multiplication. The division of the matrices as such action is not, but it can be represented as a multiplication of the first matrix to the matrix inverse to the second:A/B = A·B^(-1).
Therefore, the operation of division of matrices is reduced to two steps: finding the inverse matrix and multiply it by the first. The inverse is the matrix A^(-1), which when multiplied by A yields the identity matrix.
The formula of inverse matrix: A^(-1) = (1/∆)•B, where ∆ is the determinant of the matrix that should be nonzero. If not, then the inverse matrix does not exist. B – matrix from algebraic additions of the original matrix A.
For example, do the division of the given matrices.
Find the inverse of the second. To do this, compute its determinant and the matrix of algebraic additions. Write down the formula of the determinant of a square matrix of the third order:∆ = a11·a22·a33 + a12·a23·a31 + a21·a32·a13 – a31·a22·a13 – a12·a21·a33 – a11·a23·a32 = 27.
Define algebraic additions to these formulas:A11 = a22•a33 - a23•a32 = 1•2 – (-2)•2 = 2 + 4 = 6;A12 = -(a21•a33 - a23•a31) = -(2•2 – (-2)•1) = -(4 + 2) = -6;A13 = a21•a32 - a22•a31 = 2•2 – 1•1 = 4 – 1 = 3;A21 = -(a12•a33 - a13•a32) = -((-2)•2 - 1•2) = -(-4 - 2) = 6;A22 = a11•a33 - a13•a31 = 2•2 – 1•1 = 4 – 1 = 3;A23 = -(a11•a32 - a12•a31) = -(2•2 – (-2)•1) = -(4 + 2) = -6;A31 = a12•a23 - a13•a22 = (-2)•(-2) – 1•1 = 4 – 1 = 3;A32 = -(a11•a23 - a13•a21) = -(2•(-2) - 1•2) = -(-4 - 2) = 6;A33 = a11•a22 - a12•a21 = 2•1 – (-2)•2 = 2 + 4 = 6.
Divide the elements of the matrix algebraic additions to the value of the determinant is equal to 27. So you got the matrix inverse to the second. Now the problem is reduced to the multiplication of the first matrix to the new one.
Perform the matrix multiplication according to the formula C = A*B:c11 = a11•b11 + a12•b21 + a13•b31 = 1/3;c12 = a11•b12 + a12•b22 + a13•b23 = -2/3;c13 = a11•b13 + a12•b23 + a13•b33 = -1;c21 = a21•b11 + a22•b21 + a23•b31 = 4/9;c22 = a21•b12 + a22•b22 + a23•b23 = 2/9;c23 = a21•b13 + a22•b23 + a23•b33 = 5/9;c31 = a31•b11 + a32•b21 + a33•b31 = 7/3;c32 = a31•b12 + a32•b22 + a33•b23 = 1/3;c33 = a31•b13 + a32•b23 + a33•b33 = 0.
Advice 2 : Why can't you divide by zero
Divide by zero is impossible, every schoolboy knows that, but to many it is unclear why. The reasons for this rule can be found only at the high school, and then only if you study math. In fact, the basis of the fact that zero cannot be split, and not so complicated. To find out it would be very interesting to many students.
The reason that you can't divide by zero, lies in mathematics. While in arithmetic there are four basic operations on numbers (this is addition, subtraction, multiplication, and division) in mathematics those are only two of them (addition and multiplication). They are included in the definition of a number. To determine what is the subtraction and division, you need to use addition and multiplication, and to breed a new operation from them. To understand this point, it is useful to consider some examples. For example, an operation 10-5, with the perspective of a student of the school, means that the number 10 is subtracted, the number 5. But the math would answer the question about what's going on here, otherwise. This operation would be reduced to the equation x+5=10. The unknown in this problem is x, it is the result of the so-called subtraction. The division is is similar. It's only similarly expressed through multiplication. In this case, the result is just the right number. For example, 10:5 a mathematician would write as 5*x=10. This problem has a unique solution. Considering all this, it is possible to understand why you cannot divide by zero. Entry 10:0 would become 0*x=10. That is, the result would be a number that when multiplied by 0 gives you another number. But we all know the rule that any number multiplied by zero, gives zero. This property is included in the concept of what is zero. So it turns out that the problem of how to divide a number by zerohas no solution. This is a normal situation, many problems in mathematics have no solution. But as it may seem, this rule has one exception. Yes, no number can not divide by zero, but zero is possible? For example, 0*x=0. It's true equality. But the problem is that in the place of x can be absolutely any number. Therefore, the result of this equation would be a perfect uncertainty. There is no reason to prefer any one outcome. So zero to zero to share too. However, in the mathematical analysis of such uncertainties can handle. Check to see if there in the problem of additional conditions, which "reveal uncertainty" as it's called. But in arithmetic do not.
Advice 3 : How to calculate matrix in excel
To calculate the values of the matrix, or perform other mathematical calculations using Microsoft Office Excel. You can also use free and its analogues, the principle of operation here will be practically the same.
You will need
- - Microsoft Office Excel.
Start Microsoft Office Excel. In the data input screen you fill in this matrix for subsequent computation of its determinant. Highlight one of the unoccupied cells in the table, enter the following formula: “=MODIED(ak:fg)”. In this case, ak will denote the coordinates of the upper left corner of the matrix, and fg is the bottom right. To obtain the determinant, press Enter. The desired value will be displayed in the chosen empty cell.
Use the Excel functionality to calculate and other values. In case you do not know how to use formulas in Microsoft Office Excel, download special themed literature, and after reading you will be quite easy to navigate in this program.
Carefully read the names of the values of formulas in this software, because improper input you can spoil all the results, especially those who perform several of the same calculations for one formula at a time.
From time to time test received in Microsoft Office Excel the results of a calculation. This is due to the fact that the system could occur any change with time, in particular this applies to those who performs work on the template. Always it is useful once again to compare the results from several ongoing calculations.
Also, when working with formulas-be very careful and avoid your computer viruses. Even in case of operation with formulas in Microsoft Office Excel you need a one-time, review the functionality of this program to a greater extent, because these skills will help you in the future to better understand the automation of accounting and use Excel to perform certain tasks.