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Determinant Formula

A determinant is a word we commonly use in algebra. It is implemented in linear equations and used for many computations of matrices. Determinants are the mathematical objects which are very useful in the analysis and solution of systems of linear equations. Determinants also have many wide applications in engineering, science, and economics as well as in social science. We will study about the determinant of a matrix. In this topic, we will discuss the determinant formula with examples. Let us learn it!

Meaning of Determinant

In linear algebra, the determinant is a scalar value and is computed from the elements of the square matrix. Such a matrix obeys certain properties of transformations which are defined by the matrix. As we already know that a matrix is an array of elements or numbers. Therefore, determinant math of a matrix is the special number or value of a square matrix. Two vertical lines used on both sides are used to denote a determinant.

Determinant FormulaSource:en.wikipedia.org

Determinant in linear algebra is a useful value to provide the value of a square matrix. We denote the determinant of any matrix A by det (A), det A, or |A|.

It is a function that has an input accepts (n ×n ) matrix and output in a real number which is the determinant of the given matrix. Determinants occur throughout the many topics of mathematics. For example, many times a matrix is used to represent the coefficients in a group of linear equations. Here we use the determinant to solve these equations as efficient techniques.

Determinant Formula

  1. Let us take a matrix of order \(1 \times 1\) as:

\(\LARGE A=\begin{bmatrix} a \end{bmatrix}\).

Then its determinant will be: \(\LARGE Determinant=a\)

  1. Let us take a matrix of \(2 \times 2\) order as:

\(\LARGE A=\begin{bmatrix} p & q \\ c & d \end{bmatrix}\)

Then its determinant value will be:

\(\LARGE Determinant= pd-qc\)

  1. If the matrix is of \(3 \times 3\) order :

\(\LARGE B=\begin{bmatrix} p & q & r\\ a & b & c\\ x & y & z \end{bmatrix}\)

Then its determinant will be:

\(\LARGE Determinant =p(bz-cy)-q(az-cx)+r(ay-bx)\)

Important Properties of Determinants

  1. The value of the determinant will not change if we interchange the rows and columns.
  2. The sign of a determinant will change when we interchange any two rows or columns of a determinant with each other.
  3. The value of a determinant will be zero when any of the rows or columns of a determinant are identical to each other.
  4. For each element of a row or column which we multiply by some constant K, the value also multiplies by K.
  5. We obtain determinant as a sum of two or more determinants if we express some or all elements of a column/row as a sum of two or more terms.

Solved Examples for Determinant Formula

Q.1: Find the value of determinant of the given matrix:

\(\begin{bmatrix} 8 & 5\\ 9 & 8 \end{bmatrix}\)

Solution: Determinant will be:

Det = \(8 \times 8 – 5 \times 9\)

= 64 – 45

= 19

Q.2: Determine the determinant of the matrix:

\(\LARGE B=\begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1 \end{bmatrix}\)

Solution: Determinant will be:

= \(1\times (1 \times 1 – 1 \times 1) + 1 \times (1 \times 1 – 1 \times 1)+ 1 \times (1 \times 1 – 1 \times 1)\)

= 0

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