If A is a non-singular matrix satisfying A^3-7 A-6 I=0 then A^-1 is given by (a) (1)/(6)( A^2-7

step 1 So in the given question, we have a non -singular matrix A that satisfies the matrix equation wh

If A is a non-singular matrix satisfying A^3-7 A-6 I=0 then...

If $\mathrm{A}$ is a non-singular matrix satisfying $\mathrm{A}^{3}-7 \mathrm{~A}-6 \mathrm{I}=0$ then $\mathrm{A}^{-1}$ is given by (a) $\frac{1}{6}\left(\mathrm{~A}^{2}-7 \mathrm{I}\right)$ (b) $\mathrm{A}^{2}-7 \mathrm{I}$ (c) $\frac{1}{6}\left(\mathrm{~A}^{2}+7 \mathrm{I}\right)$ (d) $\mathrm{A}^{2}+7 \mathrm{I}$

If $\mathrm{A}$ is a non-singular matrix satisfying $\mathrm{A}^{3}-7 \mathrm{~A}-6 \mathrm{I}=0$ then $\mathrm{A}^{-1}$ is given by
(a) $\frac{1}{6}\left(\mathrm{~A}^{2}-7 \mathrm{I}\right)$
(b) $\mathrm{A}^{2}-7 \mathrm{I}$
(c) $\frac{1}{6}\left(\mathrm{~A}^{2}+7 \mathrm{I}\right)$
(d) $\mathrm{A}^{2}+7 \mathrm{I}$

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Key Concepts

Non-singular Matrix

A non-singular matrix is one that has an inverse, which also means its determinant is non-zero. This property is crucial because only matrices with non-zero determinants can be inverted, and it guarantees that any equation involving the matrix can be manipulated algebraically without running into issues related to division by zero.

Matrix Polynomial Equations

A matrix polynomial equation is an equation where a polynomial with matrix variables is set equal to the zero matrix. When a matrix satisfies such an equation, its powers and combinations with the identity matrix can be used to derive other expressions, such as the inverse. This approach leverages the algebraic structure of the polynomials and properties like distributivity and associativity in matrix operations.

Expressing the Inverse Through Polynomial Identities

In situations where a matrix satisfies a polynomial equation, it is often possible to rearrange the equation to express the inverse of the matrix as a polynomial in the matrix itself. By isolating the term containing the identity matrix and then pre-multiplying or post-multiplying by the inverse, one can derive a formula for the inverse. This method avoids direct computation of the inverse and is especially useful for matrices defined by specific algebraic identities.

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