How To Find Minor And Cofactor Of A Matrix

In this article, we will discuss how to compute the minors and cofactors of the matrices. And so, let u.s. first first with the minor of the matrix.

Minor of a Matrix

To find the minor of a matrix, nosotros accept the determinant of each smaller matrix, obtained past deleting the respective rows and columns of each chemical element in the matrix. Since in the large matrices, at that place are many rows and columns with multiple elements, therefore nosotros can make many minors of those matrices. Nosotros label these minors co-ordinate to the row and column they belong to.

We know that the square matrix has an equal number of rows and columns in it. Information technology can be of a 2x2 or 3x3 course. Each element in the foursquare matrix has its small-scale.

For case, consider the following elementary square matrix:

$B = \begin {bmatrix} m & n & o \\ p & q & r\\ s & t & u\\ \end {bmatrix}$

To find the minor of each element, we volition delete the corresponding row and column of each element and write the minors in the matrix note.

$= \begin {pmatrix} \begin {vmatrix} q & r \\ t & u\\ \end {vmatrix} & \begin {vmatrix} p & r \\ s & u\\ \end {vmatrix} & \begin {vmatrix} p & q \\ s & t\\ \end {vmatrix} \\ \begin {vmatrix} n & 0 \\ t & u\\ \end {vmatrix} & \begin {vmatrix} m & o \\ s & u\\ \end {vmatrix} & \begin {vmatrix} m & n \\ s & t\\ \end {vmatrix} \\ \begin {vmatrix} n & o \\ q & r\\ \end {vmatrix} & \begin {vmatrix} m & o\\ p & r\\ \end {vmatrix} & \begin {vmatrix} m & n \\ p & q\\ \end {vmatrix} \end {pmatrix}$

Afterwards writing the matrix in the above form, we will discover the determinant of each matrix to compute the minor of the matrix.

$M _ {11} =\begin {bmatrix} q & r \\ t & u\\ \end {bmatrix} = qu - rt$

$M _ {12} =\begin {bmatrix} p & r \\ s & u\\ \end {bmatrix} = pu - rs$

$M _ {13} =\begin {bmatrix} p & q \\ s & t\\ \end {bmatrix} = pt - qs$

$M _ {21} =\begin {bmatrix} n & 0 \\ t & u\\ \end {bmatrix}= nu - ot$

$M _ {22} =\begin {bmatrix} m & o \\ s & u\\ \end {bmatrix}= mu - so$

$M _ {23} =\begin {bmatrix} m & n \\ s & t\\ \end {bmatrix} = mt - ns$

$M _ {31} =\begin {bmatrix} n & o \\ q & r\\ \end {bmatrix} = nr - oq$

$M _ {32} =\begin {bmatrix} m & 0 \\ p & r\\ \end {bmatrix} = mr - op$

$M _ {33} =\begin {bmatrix} m & n \\ p & q\\\end {bmatrix} = mq - np$

At present, we volition solve the post-obit examples to calculate the small of the matrices.

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Let'south go

Example i

Calculate the pocket-sized of the following matrix:

$B = \begin {bmatrix} 4 & 2 & 5 \\ 6 & 3 & 1\\ 0 & 3 & 4\\ \end {bmatrix}$

Solution

Write the matrix in the following course. The below matrix is obtained by eliminating the corresponding row and column of each element.

$= \begin {pmatrix} \begin {vmatrix} 3 & 1 \\ 3 & 4\\ \end {vmatrix} & \begin {vmatrix} 6 & 1 \\ 0 & 4\\ \end {vmatrix} & \begin {vmatrix} 6 & 3 \\ 0 & 3\\ \end {vmatrix} \\ \begin {vmatrix} 2 & 5 \\ 3 & 4\\ \end {vmatrix} & \begin {vmatrix} 4 & 5 \\ 0 & 4\\ \end {vmatrix} & \begin {vmatrix} 4 & 2 \\ 0 & 3\\ \end {vmatrix} \\ \begin {vmatrix} 2 & 5 \\ 3 & 1\\ \end {vmatrix} & \begin {vmatrix} 4 & 5\\ 6 & 1\\ \end {vmatrix} & \begin {vmatrix} 4 & 2 \\ 6 & 3\\ \end {vmatrix} \end {pmatrix}$

At present, we will compute the determinant of each smaller square matrix. We know that the determinant of the square matrix $B =\begin {bmatrix} a & b \\ c & d\\ \end {bmatrix}$ is denoted by $|A|$ and calculated like this:

$|A| = ad - bc$

The resultant matrix will exist:

$= \begin {bmatrix} 9 & 24 & 18 \\ -7 & 16 & 12\\ -13 & -26 & 0\\ \end {bmatrix}$

Example 2

Calculate minors of the following matrix:

$C = \begin {bmatrix} 0 & 4 & 5 \\ 1 & 3 & 5\\ 7 & 6 & 2\\ \end {bmatrix}$

Solution

Eliminate the corresponding row and column of each element to write the matrix in the following course:

$= \begin {pmatrix} \begin {vmatrix} 3 & 5 \\ 6 & 2\\ \end {vmatrix} & \begin {vmatrix} 1 & 5 \\ 7 & 2\\ \end {vmatrix} & \begin {vmatrix} 1 & 3 \\ 7 & 6\\ \end {vmatrix} \\ \begin {vmatrix} 4 & 5 \\ 6 & 2\\ \end {vmatrix} & \begin {vmatrix} 0 & 5 \\ 7 & 2\\ \end {vmatrix} & \begin {vmatrix} 0 & 4 \\ 7 & 6\\ \end {vmatrix} \\ \begin {vmatrix} 4 & 5 \\ 3 & 5\\ \end {vmatrix} & \begin {vmatrix} 0 & 5\\ 1 & 5\\ \end {vmatrix} & \begin {vmatrix} 0 & 4 \\ 1 & 3\\ \end {vmatrix} \end {pmatrix}$

At present, discover the determinant of the foursquare smaller matrix to find the minors of all the elements in the matrix:

$B = \begin {bmatrix} -24 & -33 & -15 \\ -22 & -35 & -28\\ 5 & -5 & -4\\ \end {bmatrix}$

Cofactor

Cofactor of a matrix is related to its pocket-size. One time the modest $M_ {a,b}$ has been computed, we add together the two numbers a and b. The number obtained as a consequence of adding these two numbers is fabricated the value of the ability of -1. It is denoted as:

$(-1) ^ {a+b} \cdot M_ {a,b} = A _{a,b}$

Here, $M_ {a,b}$ is the minor and $A _{a,b}$ represents the cofactor.

Some other simpler way to understand the cofactor of a 3x3 matrix is to consider the following rule.

After finding the minor of the matrix, we modify the signs according to this rule to go the cofactor of the matrix:

$\begin {bmatrix} + & - & + \\ - & + & -\\ + & - & +\\ \end {bmatrix}$

Remember that this rule is for a 3x3 matrix.

We will calculate the cofactors of the matrices in the examples 1 and 2.

Cofactor of Example 1

In example 1, we were given the post-obit matrix:

$B = \begin {bmatrix} 4 & 2 & 5 \\ 6 & 3 & 1\\ 0 & 3 & 4\\ \end {bmatrix}$

We found its minors by eliminating the corresponding rows and columns of each element. The resultant minors of the matrix obtained was:

$= \begin {bmatrix} 9 & 24 & 18 \\ -7 & 16 & 12\\ -13 & -26 & 0\\ \end {bmatrix}$

Now, nosotros will employ this rule $\begin {bmatrix} + & - & + \\ - & + & -\\ + & - & +\\ \end {bmatrix}$ to change the sign of each element in the above matrix.

$= \begin {bmatrix} 9 & -24 & 18 \\ 7 & 16 & -12\\ -13 & 26 & 0\\ \end {bmatrix}$

The higher up matrix is the cofactor of the matrix $B =\begin {bmatrix} 4 & 2 & 5 \\ 6 & 3 & 1\\ 0 & 3 & 4\\ \end {bmatrix}$ .

Cofactor of Instance two

In instance two, we were given the following matrix:

$C = \begin {bmatrix} 0 & 4 & 5 \\ 1 & 3 & 5\\ 7 & 6 & 2\\ \end {bmatrix}$

The minors of elements obtained after eliminating the corresponding row and cavalcade of each element was below:

$B = \begin {bmatrix} -24 & -33 & -15 \\ -22 & -35 & -28\\ 5 & -5 & -4\\ \end {bmatrix}$

Now, we will apply this rule $\begin {bmatrix} + & - & + \\ - & + & -\\ + & - & +\\ \end {bmatrix}$ . Using this dominion to change signs of the elements of the matrix gives cofactors.

$B = \begin {bmatrix} -24 & 33 & -15 \\ 22 & -35 & 28\\ 5 & 5 & -4\\ \end {bmatrix}$

Hence, the above matrix is the cofactor of the matrix.

Relations of Minors and Cofactors with other Matrix Concepts

You may exist wondering what is the utilise of following this cumbersome process of finding minors and cofactors of the matrices. Well, these two concepts relate to the other concepts of the matrices. Cofactors and minors are used for ciphering of the adjoints and changed of the matrices. The adjoint of the matrix is computed by taking the transpose of the cofactors of the matrix. They also simplify the procedure of finding the determinants of the large matrices, for example, a matrix of order 4x4.