# Matrix Determinant Calculator

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## Result:

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Lets first understand, what is the determinant of a matrix and how we can find the determinant of a matrix?

Definition: A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix).

We denote matrices by capital letters. Some examples of matrices:

\[ A=\begin{bmatrix} a & b & c \\ d & e & f\end{bmatrix},B=\begin{bmatrix} g & h \\ i & j \\ k & l \end{bmatrix} \] \[ C= \begin{bmatrix} m & n & o \\ p & q & r \\ s & t & u \end{bmatrix} \]Square matrix

A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix.

\[ P= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]Determinant of a matrix

To every square matrix A = [a_{ij}] of order n, we can associate a number (real or
complex) called determinant of the square matrix A.

Determinant is denoted by | A| or det A or ∆.

\[ If\hspace{0.3cm} A= \begin{bmatrix} a & b \\ c & d \end{bmatrix},\hspace{0.3cm} \] \[then\hspace{0.3cm} |A|= \begin{vmatrix} a & b \\ c & d \end{vmatrix}=det(A) \]Example : Evaluate the determinant, \[ Δ = \begin{vmatrix} 1 & 3 & 5 \\ 6 & 7 & 9 \\ 8 & 9 & 8 \end{vmatrix}\]

Solution: Expanding along first row,

\[ Δ = 1\begin{vmatrix} 7 & 9 \\ 9 & 8 \end{vmatrix}-3\begin{vmatrix} 6 & 9 \\ 8 & 8 \end{vmatrix}+5\begin{vmatrix} 6 & 7 \\ 8 & 9 \end{vmatrix}\] $$Δ = 1(56-81)-3(48-72)$$ $$+5(54-56)$$ $$Δ = -25+72-10$$ $$Δ = 37$$Properties of Determinants

1. The value of the determinant remains unchanged if its rows and columns are interchanged.

2. If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

3. If any two rows (or columns) of a determinant are identical, then the value of determinant is zero.

4. If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

5. If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

6. If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same,

For calculating the determinant of a matrix using above matrix determinant calculator, you have to first set the order of matrix, then enter the matrix and press confirm button, after that choose the operation, then you will get the resultant matrix in the Result section.