Square matrix definition formula properties and solved examples
A square matrix is popularly known as an nxn matrix that contains an equal number of rows and columns. It is an effective way to analyse, arrange and represent data in a logical structure. The matrix is also used in mathematical equations and can provide an approximation of complicated calculations. Matrices can be of different types, and invertible matrix, symmetric matrix, singular matrix, etc. are among the important ones.
Types of Square Matrices
The following are some of the important types of square matrices.
Invertible Matrix
It is a kind of square matrix and the product of this type a matrix and its inverse is an identity matrix. The use of this invertible matrix generally is seen in the different fields of science to decrypt any coded message.
Nevertheless, a unit square matrix is another useful algebraic expression which comes from the transformation or multiplication of matrices. This unit square is basically a square with different vertices (0,0), (0,1), (1,0), (1,1).
Symmetric Matrix
A symmetric matrix is also a square matrix that follows, A\[^{T}\] = A.
Here, A\[^{T}\] is transpose f A and A\[^{-T}\] A\[^{-1}\] = I and here I stands for invertible matrix.
Singular Matrix
One of the vital features of singular matrices is that the determinant of it has to be 0. A matrix that cannot be inversed is also called a singular matrix. For example, if the element of the first column of a matrix is zero, then the determinant is zero as well.
Singular matrices are not an invertible matrix. This type of matrix works as a barrier between such matrices whose determinants are positive and whose determinants are negative. Solve and practice singular matrix example 3x3 for a better understanding of this concept. Nevertheless, besides knowing about these types of matrices, students also need to know how to find diagonal of a square.
How to Find the Diagonal of a Square Matrix?
In order to find the diagonal of a square matrix, firstly, you need to consider the first element in the 1st row and the last element in the last row. Moreover, consider all the elements that are linked in a straight way through a diagonal straight path in the matrix.
\[\begin{bmatrix} 9 & 0 & 1 & 1\\ 0 & 11 & 1 & 0\\ 1 & 1 & 4 & 1\\ 1 & 0 & 1 & 10 \end{bmatrix}\]
Here, the elements, 9, 11, 4, 10 can be joined diagonally using a straight line. Therefore, these elements are called diagonal of the square matrix.
How to Square a Matrix?
When we multiply two matrices, it is needed to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. For square matrices, students only have to multiply the elements of the two matrices to find the product. To find the square of a matrix, the matrix has to be multiplied by itself.
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FAQs on Square Matrix in Linear Algebra
1. What is a square matrix in mathematics?
A square matrix is a matrix that has the same number of rows and columns, meaning its order is n × n. For example, a matrix of order 2 × 2 or 3 × 3 is a square matrix.
- Example: A = [[1, 2], [3, 4]] is a 2 × 2 square matrix.
- The number of rows = number of columns.
- Only square matrices have determinants and inverses (if non-singular).
2. How do you find the determinant of a square matrix?
The determinant of a square matrix is a scalar value calculated using specific formulas depending on its order.
- For a 2 × 2 matrix [[a, b], [c, d]]: determinant = ad − bc.
- Example: For [[1, 2], [3, 4]], determinant = (1×4 − 2×3) = −2.
- For a 3 × 3 matrix, use cofactor expansion or the Sarrus rule.
3. What is the order of a square matrix?
The order of a square matrix is written as n × n, where n is the number of rows and columns. For example:
- A matrix with 3 rows and 3 columns has order 3 × 3.
- A matrix with 4 rows and 4 columns has order 4 × 4.
4. What is the identity matrix in a square matrix?
An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It is denoted by In.
- Example (2 × 2): [[1, 0], [0, 1]].
- Property: For any square matrix A, A × I = A.
- It acts like the number 1 in matrix multiplication.
5. What is the inverse of a square matrix?
The inverse of a square matrix is another matrix that, when multiplied by the original matrix, gives the identity matrix. For a 2 × 2 matrix [[a, b], [c, d]], the inverse is (1/(ad − bc)) [[d, −b], [−c, a]], provided ad − bc ≠ 0.
- The determinant must be non-zero.
- If determinant = 0, the matrix is singular and has no inverse.
6. What are the properties of a square matrix?
A square matrix has several important algebraic properties used in linear algebra.
- It has a determinant.
- It may have an inverse if the determinant is non-zero.
- It can be classified as diagonal, symmetric, or skew-symmetric.
- Matrix multiplication of two square matrices of the same order results in another square matrix of the same order.
7. What is a diagonal matrix in a square matrix?
A diagonal matrix is a square matrix in which all non-diagonal elements are zero.
- Example: [[3, 0], [0, 5]].
- Only elements on the main diagonal may be non-zero.
- The determinant equals the product of diagonal elements.
8. What is the difference between a square matrix and a rectangular matrix?
The main difference is that a square matrix has equal rows and columns (n × n), while a rectangular matrix has unequal rows and columns (m × n, where m ≠ n).
- Only square matrices have determinants.
- Only square matrices can have inverses.
- Example: 2 × 3 is rectangular, 3 × 3 is square.
9. How do you multiply two square matrices?
To multiply two square matrices, multiply rows of the first matrix by columns of the second matrix and add the products.
- If A and B are both n × n, then AB is also n × n.
- Element in position (i, j) = sum of products of corresponding row and column elements.
- Matrix multiplication is not commutative, meaning AB ≠ BA in general.
10. When is a square matrix called singular or non-singular?
A square matrix is called singular if its determinant is 0 and non-singular if its determinant is not 0.
- If determinant = 0, the matrix has no inverse.
- If determinant ≠ 0, the matrix has an inverse.
- This concept is important in solving systems of linear equations.
