Matrix definition types properties and how to solve problems
An array of numbers arranged in a rectangular fashion and divided between rows and columns is called a matrix in mathematics. They are usually represented by writing all the numbers contained in them within square braces. There are many types of matrices and many operations like matrix multiplication which serve as crucial topics for boards and other entrance exams.
This is one of the most vital chapters in your maths syllabus. Almost all branches of studies that derive elements from mathematics, especially computer science, use this same concept thoroughly. For example, the figure below is that of a matrix with ‘m’ horizontal rows and ‘n’ vertical columns.
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Different Types of Matrices
Column Matrix – A matrix that has elements only in one column is called a column matrix.
\[\begin{bmatrix} 1\\ 0\\ -5 \end{bmatrix}\]
Figure 2: Column Matrix
Row Matrix – A matrix that has elements only in one row is called a row matrix.
\[\begin{bmatrix} 1 & 5 & 9 \end{bmatrix}\]
Figure 3: Row Matrix
Invertible Matrix – A matrix A of size b x b is called an invertible matrix only when another matrix B exists of the same size such that AB = BA = I, where I is the identity matrix (containing only 1s in the principal diagonal) of the same dimension. In such a scenario, B is termed as the inverse matrix of A and also represented as A-1.
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Figure 4: Invertible Matrix
Singular Matrices – A matrix that has no inverse (from the previous definition) is called a singular matrix. The determinant value of the singular matrix is always 0. For example, the below matrix is singular because its determinant = 0.
For example:
\[\begin{pmatrix} 3 &12 \\ 2 & 8 \end{pmatrix}\]
The determinant is = (3 x 8) - (12 x 2)
= 24 - 24
= 0
Figure 5: Singular Matrix
Symmetric and Skew Symmetric Matrix – A matrix is called symmetric matrix if xij = xji, for all i and j, where xij = Element at ith row and jth column. Alternatively, a matrix is also called a symmetric matrix when its transpose is equal to the original matrix, AT=A. For example, the below matrix is symmetric because of the above conditions.
\[\begin{bmatrix} 3 & -2 & 4\\ -2 & 6& 2\\ 4& 2 & 3 \end{bmatrix}\]
Figure 6: Symmetric Matrix
A skew-symmetric matrix is a matrix that satisfies the condition, AT= -A.
Pop Quiz 1
A matrix is a _______ array of numbers.
Rectangular (Answer)
Square
Circular
None of the above
What is the most unique property of skew-symmetric matrices?
AT= A
AT= -A (Answer)
AT + A = I
AT. A = 0
Matrix Multiplication with a Scalar Number
A matrix can be multiplied with scalar numbers. If A = [aij]mxn (a matrix of size mxn) and k is a scalar which is to be multiplied to A, then the resultant matrix is obtained when each of the elements of A is multiplied with k, such that kA = [kaij]mxn. For example, take a look at the figure below.
k\[\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\]2x2 = \[\begin{bmatrix} ka_{11} & ka_{12}\\ ka_{21} & ka_{22} \end{bmatrix}\]2x2
Matrix Multiplication between Two Matrices
If A = [aij]m x n and B = [bij]n x p are two matrices such that the number of columns of A = number of rows of B, then the product of A and B is Cm x p. Each element cij of C is calculated with the formula below.
\[C_{ij}\] = \[\sum_{h=1}^{n}a_{ik}b_{kj}\]
Properties for Multiplying Matrices
Multiplying two matrices can only happen when the number of columns of the first matrix = number of rows of the second matrix and the dimension of the product, hence, becomes (no. of rows of first matrix x no. of columns of the second matrix).
In matrix multiplication, the order must be maintained as said in point #1. Without this order, multiplication cannot take place.
In matrix multiplication, the associative rule states that (AB)C = A(BC).
In matrix multiplication, the commutative rule states that AB ≠ BA.
Exercise
Take the following example and compute BC and A.(BC).
A = \[\begin{bmatrix} 1 & 0\\ 2 & 3\\ 3 & 1 \end{bmatrix}\] B = \[\begin{bmatrix} 1 &2 & 1& 0\\ 0 & 1 & 0 & 2 \end{bmatrix}\] C = \[\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}\]
Matrices are generally used in Geometry, but they are majorly used when the specification and representation of geometric transformation need to be done. For example, in rotations, coordinate changes and other activities. Whenever a numerical analysis is done, matrices play a vital role in its transformation. Solving computational problems is what matrices play a key role in. Matrices have a huge dimension and without them, many things might not be possible in mathematics. Other than geometry, there are other fields as well where matrices are taken into consideration.
So, this was all about matrices and all other operations and important types of them, which will be needed for your exams. To know more about other topics of mathematics, visit the Vedantu website or download the app today. We host such easy-to-read tutorials and other important guides there.
FAQs on Matrices in Maths Concepts and Applications
1. What is a matrix in mathematics?
A matrix is a rectangular arrangement of numbers, symbols, or expressions organized in rows and columns. In mathematics, matrices are used to represent data or solve systems of linear equations.
- A matrix with m rows and n columns is called an m × n matrix.
- Each number in a matrix is called an element or entry.
- Matrices are widely used in algebra, geometry, physics, computer graphics, and data science.
2. What are the different types of matrices?
The main types of matrices are classified based on their size and elements. Common types include:
- Row matrix: Only one row (1 × n).
- Column matrix: Only one column (m × 1).
- Square matrix: Same number of rows and columns (n × n).
- Zero matrix: All elements are 0.
- Identity matrix: Diagonal elements are 1, others are 0.
- Diagonal matrix: Non-diagonal elements are 0.
3. How do you add and subtract matrices?
Matrix addition and subtraction are done by adding or subtracting corresponding elements of matrices of the same order. The rule is:
- Matrices must have the same dimensions.
- Add or subtract elements in the same position.
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
A + B = [[6, 8], [10, 12]].
This operation is basic in matrix algebra.
4. How do you multiply two matrices?
Two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second matrix. If A is m × n and B is n × p, then AB is an m × p matrix.
- Multiply rows of the first matrix by columns of the second.
- Add the products to get each element.
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
AB = [[19, 22], [43, 50]].
5. What is the determinant of a matrix?
The determinant is a scalar value calculated from a square matrix that indicates whether the matrix is invertible. For a 2 × 2 matrix A = [[a, b], [c, d]], the determinant is:
det(A) = ad − bc.
- If det(A) ≠ 0, the matrix is invertible.
- If det(A) = 0, the matrix is singular (not invertible).
6. What is the inverse of a matrix and how do you find it?
The inverse of a matrix is a matrix that, when multiplied by the original matrix, gives the identity matrix. For a 2 × 2 matrix A = [[a, b], [c, d]], the inverse is:
A⁻¹ = (1 / (ad − bc)) [[d, −b], [−c, a]], provided ad − bc ≠ 0.
- Find the determinant.
- Swap diagonal elements.
- Change signs of off-diagonal elements.
- Divide by the determinant.
7. What is an identity matrix?
An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It is denoted by I and satisfies:
AI = IA = A.
- For 2 × 2: [[1, 0], [0, 1]].
- Acts like the number 1 in matrix multiplication.
8. How do you solve a system of linear equations using matrices?
A system of linear equations can be solved using matrices by writing it in the form AX = B and finding X. Steps:
- Write the coefficient matrix A.
- Write the variable matrix X.
- Write the constant matrix B.
- If A⁻¹ exists, compute X = A⁻¹B.
9. What is the transpose of a matrix?
The transpose of a matrix is formed by interchanging its rows and columns. It is denoted by Aᵀ.
- If A is m × n, then Aᵀ is n × m.
- The element in row i, column j becomes row j, column i.
10. What are the properties of matrix multiplication?
Matrix multiplication has specific algebraic properties but is not commutative. Key properties include:
- Associative: A(BC) = (AB)C.
- Distributive: A(B + C) = AB + AC.
- Not commutative: AB ≠ BA (in general).
- Identity property: AI = IA = A.
