Determinants and Matrices

Matrices Definition

Matrices and determinants are important topics for class 12th board exams, JEE, and various other competitive examinations. Our matrices and determinants notes and solved examples will help you grasp the fundamental ideas related to this chapter such as types of matrices and the definition of the determinant. The primary question that arises is What is a matrix? An ordered rectangular array of numbers, functions, symbols, or objects is called a matrix. A matrix has the order m×n if it has m rows and n columns. Such an m×n matrix are represented as:

\[\left[ {\begin{array}{*{20}{c}} {a1}& \cdots &{an} \\ \vdots & \ddots & \vdots  \\ {am}& \cdots &{amn} \end{array}} \right]\]

Types of Matrices

  1. Column matrix – A matrix having only one column - \[\left[ {\begin{array}{*{20}{c}} 3 \\  5 \\  2 \end{array}} \right]\]

  2. Row matrix – A matrix having only one row - \[\left[ {\begin{array}{*{20}{c}} 1&5&3 \end{array}} \right]\]

  3. Rectangular matrix – A matrix that has an unequal number of columns and rows -\[\left[ {\begin{array}{*{20}{c}} 6&8 \\ 0&1 \\ 3&2 \end{array}} \right]\]

  4. Square matrix – A matrix having an equal number of columns and rows - \[\left[ {\begin{array}{*{20}{c}} 5&0 \\ 3&1 \end{array}} \right]\]

  5. Diagonal matrix – A square matrix whose all elements except those in the main diagonal are zero- \[\left[ {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right],\left[ {\begin{array}{*{20}{c}}1&0&0 \\ 0&6&0 \\ 0&0&3 \end{array}} \right]\]

  6. Scalar matrix – A square matrix whose diagonal elements are all equal and all elements except those in the main diagonal are zero- \[\left[ {\begin{array}{*{20}{c}} 3&0 \\ 0&3 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 7&0&0 \\ 0&7&0 \\ 0&0&7 \end{array}} \right]\]

  7. Identity matrix – A square matrix whose main diagonal elements are ‘1’ and the other elements are all zero. It is denoted by ‘I’- \[\left[ {\begin{array}{*{20}{c}} 1&0 \\  0&1 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&0&0 \\  0&1&0 \\ 0&0&1 \end{array}} \right]\]

  8. Null matrix – A matrix of any order, all of whose elements are zero- \[\left[ {\begin{array}{*{20}{c}} 0&0 \\  0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&0&0 \\ 0&0&0 \\ 0&0&0 \end{array}} \right]\]

  9. Triangular matrix – It is a square matrix in which all the elements above or below the main diagonal are zero- \[\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 7&1&0 \\ 3&4&3 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&3&1 \\ 0&3&3 \\ 0&0&2 \end{array}} \right]\]

  10. Transpose of a matrix – The matrix obtained by interchanging the rows and columns of a matrix A and is denoted by\[{A^T} - if{\text{ }}A = \left[ {\begin{array}{*{20}{c}} 1&2&0 \\ 6&1&7 \\  5&0&3 \end{array}} \right]\] its transpose \[{A^T} = \left[ {\begin{array}{*{20}{c}} 1&6&5 \\ 2&1&0 \\ 0&7&3 \end{array}} \right]\]

Inverse of a Matrix

Inverse of a matrix usually applies for square matrices, and there exists an inverse matrix for every m×n square matrix. If the square matrix is represented by A, then its inverse is denoted by A-1, and it satisfies the property,

AA-1= A-1A = I, where I is the identity matrix. 

The determinant of the square matrix should be non-zero.

Operations on Matrices

The following operations can be performed between two or more matrices:

  1. Addition of matrix 

  2. Subtraction of matrix

  3. Multiplication of matrix

There is no division in matrices.

Solved Examples on Matrix Operations:

Addition of Matrix 

\[\begin{gathered} {\text{1}}{\text{. If  }}A = \left[ {\begin{array}{*{20}{c}}2&5&{ - 1} \\ 4&1&3 \end{array}} \right]{\text{ and }}B = \left[ {\begin{array}{*{20}{c}}  3&2&4 \\ { - 1}&5&2 \end{array}} \right],{\text{ then A + B  = }}\left[ {\begin{array}{*{20}{c}} 2&5&{ - 1} \\ { - 1}&5&2 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 3&2&4 \\  { - 1}&5&2 \end{array}} \right] \hfill \\{\text{                                                                                 }} = \left[ {\begin{array}{*{20}{c}}  {2 + 3}&{5 + 2}&{ - 1 + 4} \\   {4 - 1}&{1 + 5}&{3 + 2} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5&7&3 \\  3&6&5 \end{array}} \right] \hfill \\ \end{gathered} \]

Subtraction of Matrix 

if A = \[\begin{bmatrix}1 & 3 & -2\\ 4 & 7 & 5\end{bmatrix}\] and B = \[\begin{bmatrix}-2 & 1 & -1\\ 3 & 5 & 2\end{bmatrix}\] , then A-B = A+(-B) = \[\begin{bmatrix}1 & 3 & -2\\ 4 & 7 & 5\end{bmatrix}\] + \[\begin{bmatrix}2 & -1 & 1\\ -3 & -5 & -2\end{bmatrix}\] = \[\begin{bmatrix} 1+2 & 3-1 & -2+1\\ 4-3 & 7-5 & 5-2\end{bmatrix}\] = \[\begin{bmatrix} 3 & 2 & -1\\ 1 & 2 & 3\end{bmatrix}\]

Multiplication of Matrix

\[\begin{gathered}  3.{\text{ If }}A = \left[ {\begin{array}{*{20}{c}} 1&1 \\  0&2 \\  1&1 \end{array}} \right]{\text{ and B  =  }}\left[ {\begin{array}{*{20}{c}} 1&2 \\  2&2 \end{array}} \right],{\text{ then AB  =  }}\left[ {\begin{array}{*{20}{c}}  1&1 \\   0&2 \\  1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}}  1&2 \\   2&2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}  {1 \times 1 + 1 \times 2}&{1 \times 2 + 1 \times 2} \\   {0 \times 1 + 2 \times 2}&{0 \times 2 + 2 \times 2} \\   {1 \times 1 + 1 \times 2}&{1 \times 2 + 1 \times 2} \end{array}} \right] \hfill \\  \left[ {\begin{array}{*{20}{c}}  3&4 \\   4&4 \\   3&4 \end{array}} \right] \hfill \\ \end{gathered} \]

Determinant of a Matrix

Next, we will learn the definition of determinant of a matrix. The determinant of a matrix is defined as a scalar value that can be calculated from the elements of a square matrix. It encodes some of the properties of the linear transformation that the matrix describes and is indicated as det A, det (A), or |A|. Let us clarify it further:

A square matrix A of a specific order has a number associated with it, and this number is called the determinant of matrix A. In order for a determinant to be associated with a matrix, the latter has to be a square matrix. 

Thus, for the \[2 \times 2\] square matrix\[A = \left[ {\begin{array}{*{20}{c}}{a1}&{b1} \\   {a2}&{b2} \end{array}} \right]\], the symbol \[\left| A \right| = \left[ {\begin{array}{*{20}{c}}  {a1}&{b1} \\   {a2}&{b2} \end{array}} \right]\]

signifies a determinant of second- order. Its value is defined as: \[\left[ {\begin{array}{*{20}{c}}  {a1}&{b1} \\ {a2}&{b2} \end{array}} \right] = a1b2 - a2b1\]

Similarly, for a 3 x 3 square matrix \[A = \left[ {\begin{array}{*{20}{c}}  {a1}&{b1}&{c1} \\   {a2}&{b2}&{c2} \\   {a3}&{b3}&{c3} \end{array}} \right]\], the symbol \[\left| A \right| = \left[ {\begin{array}{*{20}{c}}  {a1}&{b1}&{c1} \\   {a2}&{b2}&{c2} \\   {a3}&{b3}&{c3} \end{array}} \right]\] 

signifies a determinant of third-order. Its value is defined as:

\[\left| A \right| = a1\left| {\begin{array}{*{20}{c}} {b2}&{c2} \\  {b3}&{c3} \end{array}} \right| - b1\left| {\begin{array}{*{20}{c}} {a2}&{c2} \\  {b3}&{c3} \end{array}} \right| + c1\left| {\begin{array}{*{20}{c}}  {a2}&{b2} \\   {a3}&{b3} \end{array}} \right|\]

Solved Example:

1. Find the determinant of matrix A, if \[A = \left[ {\begin{array}{*{20}{c}} 2&5 \\ 1&3 \end{array}} \right]\]

Solution: \[\left| A \right| = \left[ {\begin{array}{*{20}{c}} 2&5 \\  1&3 \end{array}} \right]\] (2 x 3) - (5 x 1) = 6 - 5 = 1

Properties of Determinant

Having introduced the determinant definition in math, let us go through some of the properties of determinant:

  1. If all the elements of a matrix are zero, then the determinant of the matrix is zero.

  2. For an identity matrix I of the order m×n, determinant of I, |I|= 1.

  3. If the matrix A has a transpose AT, then |AT| = |A|.

  4. If A-1 is the inverse of matrix A, then | A-1| = 1/|A| = |A|-1

  5. If two square matrices A and B are of the same size, then |AB| = |A| |B|.

  6. If c is a constant and a matrix A has the size b×b, then |cA| = cb |A|.

  7. For triangular matrices, the product of the diagonal elements gives the determinant of the matrix.

  8. Laplace’s formula: Using this formula, the determinant of a matrix is expressed in terms of its minors. If the matrix Nxy  is the minor of matrix M, obtained by eliminating the xth and yth column and has a size of ( j-1 × j-1), then the determinant of the matrix M will be given as:

|M| = ∑jy=1(−1)x+y ax,y Nx,y where (−1)x+yNx,y is the co-factor.

  1. Adjugate matrix – It is determined by transposing the matrix that contains the co-factors and is calculated using the equation:

(Adj (M))x,y = (-1)x+y Nx,y

FAQ (Frequently Asked Questions)

1. How to Memorize Derivatives of Inverse Trigonometric Functions?

In a matrix, the number of rows and columns do not necessarily have to be equal. On the other hand, in determinants, rows and columns are equal in number. When we represent a matrix, we enclose the elements within ( ), { }, or [ ]. However, while writing determinants, the elements are shown between two vertical lines, | | or ‘det’ is used before writing the elements. The next important distinction is that a matrix is just an arrangement of elements without any value, whereas a determinant has a definite value. Lastly, in the case of a matrix, if all the elements are multiplied by a constant, the matrix gets multiplied by that constant. However, for determinants, if the elements of any row or of any column are multiplied with the same constant, the original determinant will get multiplied with that constant.