If A=[ 1 0 2 0 2 3 1 3 4 ] then A^2 adj. A+A( adj. A) A+

step 1 So in the given question we have a matrix A which is given as we have a matrix A which is given

If A=[ 1 0 2 0 2 3 1 3 ...

If $A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 3 \\ 1 & 3 & 4\end{array}\right]$ then $A^{2}$ adj. $A+A($ adj. $A) A+(\operatorname{adj} A) A^{2}$ is
(a) $-15 \mathrm{I}$
(b) $-15 \mathrm{~A}$
(c) $15 \mathrm{I}$
(d) $15 \mathrm{~A}$

Key Concepts

Adjugate Matrix

The adjugate matrix (also known as the classical adjoint) is the transpose of the cofactor matrix. It is particularly important because it satisfies the relation A · adj(A) = det(A) I, which provides a link between a matrix, its determinant, and its inverse. This property often helps simplify expressions in matrix algebra that involve the adjugate.

Matrix Determinant

The determinant is a scalar attribute of a square matrix that encodes properties such as invertibility and volume scaling. In the context of the adjugate, the determinant plays a crucial role via the formula A · adj(A) = det(A) I. This relation is used to simplify complex matrix expressions and is foundational in understanding matrix inverses.

Matrix Multiplication and Associativity

Understanding the rules of matrix multiplication and the associative property is key when dealing with expressions involving powers of matrices and products with the adjugate. Since matrix multiplication is associative, grouping factors can simplify calculations, especially when exploiting identities like A · adj(A) = det(A) I.

Identity Matrix and its Role

The identity matrix, which has ones on the diagonal and zeros elsewhere, serves as the multiplicative identity in matrix algebra. It plays a significant role in many matrix identities, including A · adj(A) = det(A) I, where the identity matrix facilitates the conversion of matrix products into scalar multiples and simplifies expressions involving matrix powers and adjugates.

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