If A=[ a b c -a ] and A^2 is the identity...

If $A=\left[\begin{array}{cc}a & b \\ c & -a\end{array}\right]$ and $A^{2}$ is the identity matrix of order 2, then the relation between $a, b$ and $c$ is
(a) $\mathrm{a}^{2}=\mathrm{bc}$
(b) $1-a^{2}-b c=0$
(c) $1-a^{2}+b c=0$
(d) $1+a^{2}-b c=0$

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Key Concepts

Matrix Multiplication

Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second matrix. This fundamental operation in linear algebra is used to compute powers of matrices, where each entry in the resulting matrix is determined by a sum of products of entries from the original matrices.

Identity Matrix

The identity matrix is a square matrix with ones on its main diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix multiplication, meaning that when any matrix is multiplied by the identity matrix, it remains unchanged.

Equating Matrix Entries

When two matrices are equal, every corresponding entry must be the same. This principle is often applied to set up equations by comparing the entries on both sides of a matrix equation, enabling one to derive relationships among the variables in the matrix.

Structured 2x2 Matrices

Structured 2x2 matrices, which have a specific arrangement of entries, can simplify calculations such as finding powers of the matrix. Recognizing the structure aids in deducing inherent relationships among the elements, as demonstrated by equating the resulting expressions from matrix operations to those of a standard form like the identity matrix.

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