Difference between revisions of "Manuals/calci/MINVERSE"

Revision as of 05:06, 1 January 2014

MINVERSE(arr)

This function gives the inverse matrix for the given matrix.
We have to find a inverse of a matrix then it should satisfy the following conditions
1.A matrix must be a square matrix.
2.It's determinant not equal to 0.
Let $A$ be the 2x2 matrix with the elements

$A={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}$ .

A^{-1}={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}^{-1}={\frac {1}{ad-bc}}*{\begin{bmatrix}d&-b\\-c&a\\\end{bmatrix}}

Now let $A$ be the matrix is of order $nXn$ .
Then the inverse of $A$ is <math>A^{-1}= \frac{1}{det(A)}*adj(A)<math>
Where <math>adj(A)<math> is the adjoint of <math>A<math>.
Adjoint is the matrix formed by taking the Transpose of the Co-factor matrix of the original matrix.
Also <math>A.A^-1=A^-1.A=I<math>, where <math>I<math> is the identity matrix.
Non-square matrices do not have inverses.
Not all square matrices have inverses.
A square matrix which has an inverse is called invertible or non-singular, and a square matrix without an inverse is called non-invertible or singular.
This function will return the result as error when

1. Any one of the cell is non-numeric or any cell is empty or contain text
2. Suppose number of rows not equal to number of columns

Matrix A A=(4 3

  3     2)

MINVERSE(B5:C6)=(-2 3

                 3       -4)

MATRIX A A=(3 4

     6       8)

MINVERSE(C4:D5)=Null, because its det value is 0. MATRIX A A=(2 3

  4     7)

MINVERSE(B4:C5)=(3.5 -1.5

               -2           1)

@@ Line 12: / Line 12: @@
 c & d \\
 \end{bmatrix}</math>.
-*Then the inverse of matrix <math>A</math> is denoted by <math>A^{-1}</math>.So <math>A^{-1}=\begin{bmatrix}
+*Then the inverse of matrix <math>A</math> is denoted by <math>A^{-1}</math>.
+:<math>A^{-1}=\begin{bmatrix}
 a & b \\
 c & d \\
-\end{bmatrix}^{-1}= \frac{1}{ad-bc} *   \begin{bmatrix}
+\end{bmatrix}^{-1}= \frac{1}{ad-bc} *  \begin{bmatrix}
 d & -b \\
 -c & a \\
-\end{bmatrix} </math>
+\end{bmatrix}
 </math>
-*Now let A be the matrix is of order nXn.
+*Now let <math>A</math> be the matrix is of order <math>nXn</math>.
-*Then the inverse of A is A^-1= 1/det(A) . adj(A)
+*Then the inverse of <math>A</math> is <math>A^{-1}= \frac{1}{det(A)}*adj(A)<math>
-*Where adj(A) is the adjoint of A.
+*Where <math>adj(A)<math> is the adjoint of <math>A<math>.
-*Adjoint is the matrix formed by taking the transpose of the co-factor matrix of a given original matrix.
+*Adjoint is the matrix formed by taking the Transpose of the Co-factor matrix of the original matrix.
-*Also A.A^-1=A^-1.A=I, where I is the identity matrix.Non-square matrices do not have inverses.
+*Also <math>A.A^-1=A^-1.A=I<math>, where <math>I<math> is the identity matrix.
+*Non-square matrices do not have inverses.
 *Not all square matrices have inverses.
 *A square matrix which has an inverse is called invertible or non-singular, and a square matrix without an inverse is called non-invertible or singular.