Q.Find the inverse of $A=\begin{bmatrix}3&2\\1&1\end{bmatrix}$ using the adjoint-determinant method.

Two square matrices are inverses of each other exactly when both products give the identity matrix.

By definition, if $A$ and $B$ are square matrices of the same order such that $AB=BA=I$ (the identity matrix), then $B$ is called the inverse of $A$ (written $B=A^{-1}$) and $A$ is the inverse of $B$. A single-sided product, or a product equal to the zero matrix, does not establish an inverse relationship.

A matrix has no inverse exactly when its determinant is zero.

A square matrix is invertible only if it is non-singular, i.e. $\det\ne 0$. "No inverse matrix" means the matrix IS singular, so we need $\det = 0$.

$\det\begin{pmatrix}4 & -k\\ -2 & 3\end{pmatrix} = 4(3) - (-k)(-2) = 12 - 2k$

Set this to zero: $12-2k=0 \Rightarrow k=6$.

By definition, A and B are inverses of each other exactly when their product (in either order) gives the identity matrix.

A square matrix $B$ is the inverse of $A$ if $AB=BA=I$, where $I$ is the identity matrix of the same order. This is the standard definition of matrix invertibility - both products must equal the identity, not zero or any other matrix.

Inverse of a matrix is defined by the identity relation on both sides.

By definition, if $A$ is a square matrix and there exists a matrix $B$ of the same order such that $AB=BA=I$ (the identity matrix), then $B$ is called the inverse of $A$ (written $B=A^{-1}$), and $A$ is the inverse of $B$. Both products must equal the identity matrix, not zero or in only one order.