Determinant of a square matrix

The determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle

To compute the inverse of a matrix, the determinant is required. For each square matrix A, there is a unit scalar value known as the determinant of A, denoted by det A or |A|.If det(A)=0, the matrix is said to be singular.The determinant contains the same elements as the matrix which are enclosed between vertical bars instead of brackets in a scalar equation. If A = [A] is a single element (1×1), then the determinant is defined as the value of the element

Then |A| =det A= a₁₁

If A is (n x n), its determinant may be defined in terms of the order (n-1) or less.

The determinant of A = $\lbrack \begin{matrix}
3 & -2 & \\
17 & 11 & \\
\end{matrix}
\rbrack $ is written as $\vert \begin{matrix}
3 & -2 & \\
17 & 11 & \\
\end{matrix}
\vert $. For second-order determinants we use the following formula:

If A is an n x n matrix and one row and one column are deleted, the resulting matrix is an (n-1) x (n-1) submatrix of A.The determinant of such a submatrix is called a minor of A and is designated by m_ij, where i and j correspond to the deleted row and column, respectively. This means that m_ij is the minor of the element a_ij in A.

The cofactor C_ij of an element a_ij is defined as:

When the sum of a row number i and column j is even, c_ij = m_ij and when i+j is odd, c_ij =-m_ij

The determinant of an n x n matrix A can be written as:

The determinant of Ais essentially the sum of the products of the elements of the first row of A and their corresponding cofactors.

For using minors to calculate third-order determinants, each member of one row is multiplied by its minor. The signs of the product’s alternate, beginning with the second product. The definition below shows an expansion using the elements in the first for of the determinant. However, ANY row can be used.

Example: Evaluate the determinant of A = $\lbrack \begin{matrix}
2 & 3 & 4 & \\
6 & 5 & 7 & \\
1 & 2 & 8 & \\
\end{matrix}
\rbrack $

Example: Find the determinant

The cofactors of the first row are:

There is another simpler method utilizing diagonals for evaluating a third-order determinant.

STEP 1: Write the first two columns on the right side of the determinant as shown below.
STEP 2: Draw a diagonal from each element in the top row downwards and deduce the product of the numbers on each diagonal.
STEP 3: Now draw a diagonal from each element in the bottom row upwards and deduce the product of the numbers on each.
To find the value of the determinant, add the products in the first set of diagonals, and then subtract the products from the second set of diagonals.
The value is:aei + bfg + cdh – gec – hfa – idb

Determinants are utilized to calculate the area of a triangle when you know the coordinates of the three vertices. The area of a triangle whose vertices have coordinates (a, b), (c, d), (e, f) can be found by using the formula:

Example: Find the area of the triangle whose vertices have coordinates (-4, -1), (3, 2), (4, 6).

Assign values to a, b, c, d, e, and f in the above formula: a = -4, b = -1, c = 3, d = 2, e = 4, f = 6

Now use the diagonal method and add the bottom products and subtract the top products: -8 + (-4) + 18 – 8 – (-24) –(-3) = 25. The value of the determinant is 25. And when we apply that to the area formula, we obtain ½ (25) = 12.5. The area of the triangle is 12.5 square units.

When one or more elements of a determinant are unknown, but the value of the determinant is known then we can use expansion to find the values of the variable.

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