21-06-2024 07:42

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Tags : mathematics Row echelon form Rank of matrix linear algebra

Minor method to find rank

The Matrix has to be non zero

Rank will be R if the matrix has

• The Minors of order R is non zero
• Every minor with order greater than R is zero

Certainly! Let’s calculate the rank of an example matrix using the minor method. We’ll follow these steps:

1. Calculate the determinant of the matrix (let’s call it ($A$)).
2. If $(\det (A)\ne 0)$, then the rank of matrix ($A$) is equal to the order of matrix ($A$).
3. If (\det(A) = 0), then the rank of the matrix is equal to the order of the maximum possible nonzero minor of the matrix.

Now, let’s consider an example matrix:

$A=\left(\begin{array}{ccc}2& 1& 1\\ 5& 3& 4\\ 7& 4& 5\end{array}\right)$

$\mathrm{Solve}\frac{d^{4}y}{d{x}^4}-13\frac{d^{2}y}{d{x}^2}+36y=0..$ First, we’ll calculate the determinant of matrix (A):

$\det (A)=\left|\begin{array}{ccc}2& 1& 1\\ 5& 3& 4\\ 7& 4& 5\end{array}\right|$

Using cofactor expansion, we get:

$\det (A)=2\cdot \left|\begin{array}{cc}3& 4\\ 4& 5\end{array}\right|-1\cdot \left|\begin{array}{cc}5& 4\\ 7& 5\end{array}\right|+1\cdot \left|\begin{array}{cc}5& 3\\ 7& 4\end{array}\right|$

Calculating the determinants of the 2x2 submatrices:

\begin{align*} \begin{vmatrix} 3 & 4 \\ 4 & 5 \end{vmatrix} &= 3 \cdot 5 - 4 \cdot 4 = 1 \\ \begin{vmatrix} 5 & 4 \\ 7 & 5 \end{vmatrix} &= 5 \cdot 5 - 4 \cdot 7 = -3 \\ \begin{vmatrix} 5 & 3 \\ 7 & 4 \end{vmatrix} &= 5 \cdot 4 - 3 \cdot 7 = -1 \end{align*}

Now, compute the determinant of (A):

$\det (A)=2\cdot 1-1\cdot (-3)+1\cdot (-1)=6$

Since $(\det (A)\ne 0)$, the rank of matrix ($A$) is equal to its order, which is 3.

Therefore, the rank of $matrix(A)$ using the minor method is 3.

Feel free to ask if you have any more questions or need further assistance! 😊

References