Algorithm Explanation:

This algorithm is aimed at transforming a matrix $A$ into a modified version where parts of the matrix are replaced by specific elements from the inverse of a minor, while other elements are set to zero. Let’s break it down step-by-step with an example:

Step 1: Identify a Non-Singular Minor of Order $r$

• Non-Singular Minor: A minor is a square submatrix obtained from a larger matrix. A non-singular minor is a submatrix with a non-zero determinant.
• Rank $r$: The rank of a matrix is the number of linearly independent rows (or columns). For a matrix $A_{m \times n}$, if its rank is $r$, then there exists at least one minor of order $r$ (i.e., an $r \times r$ submatrix) that has a non-zero determinant.

Step 2: Calculate the Inverse of this Minor

• Inverse: Once the non-singular minor $M$ is identified, calculate the inverse of this minor, denoted as $M^{-1}$.
• Transpose: Then, compute the transpose of this inverse, denoted $(M^{-1})^T$. This means flipping the matrix over its diagonal, turning rows into columns and vice versa.

Step 3: Replace Elements in the Original Matrix $A$

• Replacement in $A$: The elements of $A$ that correspond to the positions of $M$ are replaced by the corresponding elements of $(M^{-1})^T$.
• Zeros Elsewhere: All other elements of the matrix $A$ (i.e., those not part of the minor $M$) are replaced with zeros.

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Example:

Let’s consider a simple example with a $3 \times 3$ matrix $A$, and assume its rank $r$ is 2.

Suppose we have:

1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$$ #### Step 1: Find a Non-Singular Minor of Order 2 Choose a submatrix of order 2 that has a non-zero determinant. Let’s pick the top-left $$2 \times 2$$ minor: $$M = \begin{pmatrix} 1 & 2 \\ 4 & 5 \end{pmatrix}$$ The determinant of $$M$$ is: $$\text{det}(M) = 1(5) - 2(4) = 5 - 8 = -3 \quad (\text{Non-zero, so it is non-singular})$$ #### Step 2: Calculate $$(M^{-1})^T$$ First, calculate the inverse of $$M$$: \[ M^{-1} = \frac{1}{\text{det}(M)} \begin{pmatrix} 5 & -2 \\ -4 & 1 \end{pmatrix} = \frac{1}{-3} \begin{pmatrix