Matrices and Determinants - Matrices and Determinants

An orderly arrangement of numbers or functions in a rectangular form is defined as a matrix. A matrix is denoted by capital letters. The numbers or functions are called elements and are distinguished by its position in rows and columns. The elements are represented by aij, where ‘i’ denotes the corresponding row, and ‘j’ denotes the corresponding column.

Let A be a matrix, A=

In the above example above, a11=1, a12= 2, a13=3,a21=4 and so on. 

Order of the matrix is denoted as m n, where m is the number of rows and n is the number of columns a matrix has. The number of rows multiplied by the number of columns gives the number of elements, i.e., number of elements=m x n. In the above example above, an order of the matrix = 3x3 and nNumber of elements=3x3=9.

Properties ( when A and B are two matrices)

• AB ≠ BA
• (AB)C = A(BC)
• A.(B+C) = A.B + A.C

For every square matrix, a number can be computed (real/complex), called the particular matrix’s determinant.

For instance, consider a matrix of order 2, A=

For a matrix of order 3, a determinant can be calculated by expanding a determinant along a row or column. Hence, there are six ways to compute a determinant of order 3, along the three rows or [A] along with the three columns. Let us understand it better using an example.

Properties

• The values of a determinant remain unchanged even after interchanging its rows and columns.
• The sign of a determinant changes if any two rows or columns are interchanged.
• The value of a determinant in which any two rows are identical is zero.
• If each element of a row or column is multiplied by a constant, then the value gets multiplied by the same constant.
• If some or all elements of a row or column are expressed as the sum of two or more items, then the determinant can be defined as the sum of two or more determinants.
• If equimultiples of corresponding elements are added to each element of a row or column, then the determinant’s value remains the same.

The fundamentals of matrix and determinants are introduced to students in grade 12. The CBSE Board gives a weightage of 12 marks for the full unit of Algebra, under which these topics are discussed. 

1. In the matrix A= 

Solution:

The order of the matrix

The elements a13, a21,  a33, a24, a23

The number of rows is 3, and the number of columns is 4. Order= 34.a13=19,a21=35,a33= - 5,a24=12,a23= 5/2.

2. Evaluate the determinant 

Solution:

= 2x(-1) - (-5) x 4 = - 2 + 20= 18. 

3. Evaluate the determinant 

Solution:

= (cosθ x cosθ) - (sinθ x - sinθ)

= cos2θ + sin2θ

=1

Q: What are Name the different types of matrices?.

Q: When are two matrices equal?

Q: Does every matrix have its corresponding determinant?

Q: What is a determinant of the matrix of order 1?

Q: What is the determinant’s value if all the row or column elements are zeroes?

• Column matrix
• Row matrix
• Square matrix
• Diagonal matrix
• Scalar matrix
• Identity matrix
• Zero matrix