Key Points

For a square matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is calculated as $|A| = ad - bc$. It is a scalar value associated with the matrix.

For a matrix $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$, the determinant can be found by expanding along the first row: $|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$. The same value is obtained by expanding along any row or column.

A square matrix A is called singular if its determinant is zero, that is $|A| = 0$. If $|A| \neq 0$, the matrix is called non-singular. Only non-singular matrices have an inverse.

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by the formula $\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$. Since area is a positive quantity, we take the absolute value of the determinant.

Three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear if and only if the area of the triangle formed by them is zero. This means $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0$.

The minor $M_{ij}$ of an element $a_{ij}$ is the determinant of the sub-matrix obtained by deleting the $i$-th row and $j$-th column. The cofactor is defined as $A_{ij} = (-1)^{i+j} M_{ij}$.

The value of a determinant is the sum of the products of the elements of any single row or column with their corresponding cofactors. For example, expanding along row 1 gives $|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$.

If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero. For example, $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0$.

The adjoint of a square matrix A, denoted by adj(A), is the transpose of the matrix formed by the cofactors of the elements of A. If $C$ is the matrix of cofactors, then $\text{adj}(A) = C^T$.

For any square matrix A of order $n$, the following relationship holds: $A(\text{adj } A) = (\text{adj } A)A = |A|I$, where $I$ is the identity matrix of order $n$.

A square matrix A is invertible (has an inverse) if and only if it is non-singular ($|A| \neq 0$). The inverse is given by the formula $A^{-1} = \frac{1}{|A|} (\text{adj } A)$.

For any non-singular square matrix A of order $n$, the determinant of its adjoint is given by the property $|\text{adj}(A)| = |A|^{n-1}$.

A system of linear equations like $a_1x+b_1y+c_1z=d_1$, etc., can be written in matrix form as $AX = B$. If A is non-singular, the system has a unique solution given by $X = A^{-1}B$.

For a system $AX=B$: 1) If $|A| \neq 0$, it is consistent with a unique solution. 2) If $|A|=0$ and $(\text{adj } A)B \neq O$, it is inconsistent (no solution). 3) If $|A|=0$ and $(\text{adj } A)B = O$, it may be consistent (infinitely many solutions) or inconsistent.

For any two square matrices A and B of the same order, the determinant of their product is the product of their individual determinants. That is, $|AB| = |A||B|$.

If A and B are invertible matrices of the same order, then their product AB is also invertible. The inverse of the product is given by the reversal law: $(AB)^{-1} = B^{-1}A^{-1}$.

If A is a square matrix of order $n$ and $k$ is any scalar, then the determinant of the matrix $kA$ is given by the property $|kA| = k^n|A|$.