गणित एवं संख्यात्मक विश्लेषण

Step-by-step matrix inversion:

• Erroneous adjoint calculation.
• Determinant $\det(A) = 0(4 - 6) - 2(2 - 6) + 4(6 - 12) = 0 + 8 - 24 = -16$. The cofactor matrix is: $C = \begin{pmatrix} -2 & 4 & -6 \\ 10 & -12 & 6 \\ -12 & 8 & -4 \end{pmatrix}$. Transposing to get adjoint: $\operatorname{adj}(A) = \begin{pmatrix} -2 & 10 & -12 \\ 4 & -12 & 8 \\ -6 & 6 & -4 \end{pmatrix}$. Dividing by $-16$ gives $A^{-1} = \begin{pmatrix} 1/8 & -5/8 & 3/4 \\ -1/4 & 3/4 & -1/2 \\ 3/8 & -3/8 & 1/4 \end{pmatrix}$.
• Has $3/8$ instead of $3/4$ at position (2,2).
• Distractor with zeroes.

Axiomatic rank properties of matrices:

• An invertible matrix has full rank equal to its order $n$, never $0$.
• The $n \times n$ identity matrix has rank $n$.
• non-singular square matrix has rank $n \gt 0$.
• The rank of a matrix is defined as the maximum number of linearly independent row/column vectors. The rank of a matrix is strictly greater than or equal to $1$ for any non-zero matrix. The only matrix whose rank is identically $0$ is the zero matrix (null matrix).

Conditions for invertibility of square matrices:

• singular matrix has determinant equal to zero by definition.
• non-singular matrix has non-zero determinant.
• By the Fundamental Invertible Matrix Theorem, a square matrix $A$ is invertible if and only if its determinant is non-zero ($\det(A) \ne 0$). Hence, this statement is always true.
• non-invertible matrix has determinant equal to zero.