Before discussing the operations of the matrix, let’s discuss what a matrix is.
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A matrix is a rectangular (2D) array of numbers or symbols which are generally arranged in rows and columns. One can think of it as a table of numbers.
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The order of the matrix is defined as the number of rows and columns.
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The entries are the numbers in the matrix and each number is known as an element.
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The plural of matrix is matrices.
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The size of a matrix is referred to as ‘n by m’ matrix and is written as m×n, where n is the number of rows and m is the number of columns.
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For example, we have a 3×2 matrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.
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The dimensions of a matrix can be defined as the number of rows and columns of the matrix in that order. Since matrix A given above has 2 rows and 3 columns, it is known as a 2×3 matrix.
What are the Different Types of Matrix?
There are different types of matrices. Here they are –
1) Row matrix
2) Column matrix
3) Null matrix
4) Square matrix
5) Diagonal matrix
6) Upper triangular matrix
7) Lower triangular matrix
8) Symmetric matrix
9) Anti-symmetric matrix
Adding Matrices
Two matrices must have an equal number of columns and rows in order to be added. The sum of any two matrices suppose A and B will be a matrix which has the same number of rows and columns as do the matrices A and B. The sum of A and B, can be denoted as A + B, is computed by adding corresponding elements of A and B.
A + B = [begin{bmatrix} a_{11} & a_{12} & cdots & a_{1n}\ a_{21} & a_{22} & cdots & a_{2n}\ vdots & vdots & ddots & vdots\ a_{m1} & a_{m2} & cdots & a_{mn} end {bmatrix}] + [begin{bmatrix} b_{11} & b_{12} & cdots & b_{1n}\ b_{21} & b_{22} & cdots & b_{2n}\ vdots & vdots & ddots & vdots\ b_{m1} & b_{m2} & cdots & b_{mn} end {bmatrix}]
= [begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & cdots & a_{1n} + b_{1n}\ a_{21} +b_{21} & a_{22} + b_{21}& cdots & a_{2n} + b_{2n}\ vdots & vdots & ddots & vdots\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & cdots & a_{mn} + b_{mn} end {bmatrix}]
Matrix Sums and Answers
Let us suppose that we have two matrices A and B.
Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In order words, you can add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 matrix.
A = [begin{bmatrix} 1 & 2 & 3\ 7 & 8 & 9end {bmatrix}] B = [begin{bmatrix} 5 & 6 & 7\ 3 & 4 & 5end {bmatrix}]
A + B = [begin{bmatrix} 1 + 5 & 2 + 6 & 3+ 7\ 7 + 3& 8 + 4 & 9 + 5end {bmatrix}]
A + B = [begin{bmatrix} 6 & 8 & 10\ 10 & 12 & 14end {bmatrix}]
Note: Keep in mind that the order in which matrices are added is not important; thus, we can say that A + B = B + A.
Properties of Matrix Addition
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1. The Commutative Law |
If matrix A = [aij] and matrix B = [bij] are the matrices of the same order, we can say m × n, then A + B will be equal to B + A. |
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2. The Associative Law |
For any three matrices namely A , B, and C, A = [aij], B = [bij], and C = [cij] of the same order, say suppose m × n, then we can write (A + B) + C is equal to A + (B + C). |
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3. The Existence of Additive Identity |
Let us say we have a matrix A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O is equal to O + A = A. In simpler words, we can say that O is the additive identity for matrix addition. |
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4. The Existence of Additive Inverse |
Let matrix A = [aij]m×n be any matrix, then we have another matrix as – A = [–aij]m×n such that A + (–A) is equal to (–A) + A= O. So, – A can be known as the additive inverse of A or negative of A. |
Applications of Matrices
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Matrix is used in many branches of mathematics, for example, for calculations related to vectors like finding the derivative, integration, the integral of a matrix, etc.
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Matrices are widely used in matrix and linear algebra, in particular, to represent and solve linear systems of equations.
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A matrix is also used in solving eigenvalue problems, symmetric and eigenvectors, linear regression, optimization problems, etc.
Solved Example Problems
Question 1. Add the following matrices:
[A = begin{bmatrix}1 & 2&3 \7 & 8&9 end{bmatrix}] [B =begin{bmatrix}5 & 6&7 \3& 4&5 end{bmatrix} ]
Solution: We have two matrices A and B.
Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In other words, you can add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 matrix.
[A = begin{bmatrix}1 & 2&3 \7 & 8&9 end{bmatrix}] [B =begin{bmatrix}5 & 6&7 \3& 4&5 end{bmatrix} ]
[ A + B =begin{bmatrix}1+5 & 2+6&3+7 \7+3& 8+4&9+5 end{bmatrix} ]
[ A+B = begin{bmatrix}6 & 8&10 \10 & 12&14 end{bmatrix} ]
Question 2. Add the following matrices.
[ A = begin{bmatrix}3 & 4&9 \12& 11&35 end{bmatrix}] [ B = begin{bmatrix}6 & 2 \5 & 8 end{bmatrix} ]
Solution: Let’s add the following two matrices A and B. As we know that matrices are added entry-wise, we have to a
dd the 3 and the 6, the 12 and 5, the 4 and the 6, and the 11 and the 8. But what do I add to the entries 9 and 35? There are no corresponding entries in the second matrix that can be added to these entries in the first matrix. So here’s the answer:
We can’t add these matrices A and B, because these matrices are not of the same size.
Question 3. Suppose X, Y, Z, W, and P are matrices of the given order 2 × n, 3 × k, 2 × p, n × 3, and p × k, respectively. The restriction on n, k, and p so that PY + WY can be defined as-
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k is arbitrary, p = 2
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p is arbitrary, k = 3
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k = 2, p = 3
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k = 3, p = n
Solution: In this, the order of matrix P = p × k, order of W = n × 3, order of matrix Y = 3 × k. Thus, the order of PY = p×k, when k is equal to 3. And the order of WY = p × k, where p = n. Thus, option (D).