Matrix is a key concept in mathematics and among the most powerful tools, with numerous applications such as solving linear equations, financial planning, sales projection, cost analysis, etc. Matrix for 12th class covers important matrix concepts such as kinds, order, composite elementary transition operations, etc. Here, students can find the detailed elaboration of matrix concepts. Matrices for the CBSE class aid students in their higher education by covering all fundamental topics. To get good grades in the exams, go over notations on class 12 matrices.
|Table of Contents|
|Types of Matrices|
|What is Equality of Matrices?|
|Matrix Multiplication Characteristics|
A Matrix is a collection of functions composed of a definite order rectangular series of integers. The array’s numbers are the matrix’s entities or elements. The lateral array of components in the mixture is referred to as the columns, and the vertical variety of components is referred to as the columns. When a matrix does have m rows and n columns, it is referred to as a composite of sequence m x n. Matrices are rectangular arrays of figures or functions. It is two-dimensional because it is a rectangle array. A two-dimensional matrix is made up of several rows (m) and columns (n) (n). The order of the matrix is m x n.
Types of Matrices
There are different types of matrices depending upon their elements and order:
- Column matrix
- Row matrix
- Square matrix
- Diagonal matrix
- Scalar matrix
- Identity matrix
- Zero matrix
Column Matrix: A column matrix is a matrix with only one column. If n = 1, A = [aij]mxn would be a section matrix. As a result, the valuation of a data matrix is 1. As a result, the sequence is m. This is also known as a column vector.
Row matrix: A row matrix is a matrix of process m x n, in which m is the total of rows and n is the number of columns, if and only if m = 1. As a result, a matrix of order 1 x n is a row structure. The diagonal lines of elements form a row matrix. A row data structure is a data arrangement of elements that are ordered horizontally.
Square matrix: Square matrix is one with the same number of columns. In mathematics, the m m matrix is the matrix form of order metres. If we amplify and add any 2 square matrices, the resulting matrix retains its original order. The system . This means 2 is a two-row, two-column matrix. A square matrix of 3 has three rows and three columns, so its order is three. There are an equal number of columns and rows. The sequence of a matrix is still the sum of all of the diagonal cells of a square matrix. An identity matrix is one in which all of the diagonal cells of a matrix form are equal to 1. Various operations on a square matrix can be performed, such as inverse. Only the square matrix allows for the calculation of the determinant value. A square matrix’s order of transpose is like the original matrix’s.
Diagonal matrix: A Diagonal Matrix is a matrix wherein every element except for the primary diagonal elements is zero. When I is not equivalent to j, applying formulas, D = n x n is called angled modelling if dij = 0. A diagonal matrix would be a square matrix B = m m if all non-diagonal matrices are zero; a diagonal matrix is a matrix B = [bij] m m if bij = 0 when I j.
Scalar matrix: A matrix, say A = [aij]n n, is referred to as a scalar matrix and = 0, once I = j, and aij = k, when I = j. The dimension of the linear function matrix contains only identical scalar elements. The scalar matrix has an order of n x n. As a result, it has the same number of rows. As a result, it also is a Town centre matrix.
Identity matrix: An identity square matrix where all principal diagonal elements are one and other elements are zero. The letter “In” or simply “I.” The result of multiplying any matrix by the identity matrix is a given matrix. The given matrix’s elements remain unchanged. In other words, an identity matrix is one in which all of the main diagonals of a matrix form are 1’s, and the rest are o’s. A Square Matrix is always used. The matrix itself is computed by adding any sequence to the unit matrix. After simply multiplying inverse matrices, get an identity.
Zero matrices: A zero matrix is a set of zero elements organised into columns and rows. A 0 matrix has all of its submissions equal to 0. It is represented by the letter ‘O,’ which can be conveyed with such a substring to represent the matrix’s dimension.
What is Equality of Matrices?
Equality of Matrices: While two or more matrices are equal, this is called matrix equality. Matrices are equal if they have the same number of rows and columns and the same number of corresponding elements. Equality of matrices does not hold for either of the properties mentioned above. This means that if the matrices’ order is not equal, or at least each pair of the corresponding pixel is not equivalent, the two vectors were said to have been unequal.
Basic operations just on a matrix are addition, subtraction, and multiplication. When adding or subtracting matrices, they must be in the same order, and when multiplying, the column in the first matrix must equal the row in the matrix.
- Addition of Matrices
- Subtraction of matrices
- Matrix multiplication
Addition of Matrices: To add matrices, they must have the same aspects, i.e. the matrix must be in the same order. If A[aij]mxn and B[bij]mxn seem to be two matrices of the same order, their total amount A plus B is a structure, so each component is the total amount of the elements. In other words, A plus B = [aij plus 1 bij]mxn
Subtraction of matrices: Subtraction of vectors or array subtraction is only possible if both matrices have the same number of rows. We deduct the components in each row or column from the corresponding pixel in the other matrix’s row and column when subtracting the matrix.
Example: Using the different methods of matrix description, find the component in the first row and 3rd column of a matrix B – A. If a13 = 14 would be a component, Or b13 = three years later is an aspect in B.
Solution: To find the element in the first rack and third column of a matrix B – A, we must compute b13 – a13 using the matrix subtraction definition.
b13 minus a13 = -3 minus 14 = -17
The element of B – A’s first row and the third section is -17.
Matrix multiplication: The outcome is a matrix when two equations are multiplied. The multiplier is a rational function for whom output is a matrix. Matrix multiplication is only possible in linear algebra when the matrix is consistent. In general, unlike arithmetic multiplication, matrix multiplication is not commutative, which means that the multiplication of composite A and B, denoted as AB, can indeed be equal to BA, i.e., AB BA. As a result, the order of multiplying is critical when multiplying matrices.
Matrix Multiplication Characteristics
There are some mathematical properties of composite multiplications in linear algebra. These characteristics of Matrix Multiplication are as follows:
- Non-Commutative: Multiplier is non-commutative, which means that when two matrices, A and B, are multiplied, the result is AB BA.
- Distributivity: When multiplying matrices, the socially beneficial property can be used — in other words, A(B + C) = AB + BC, provided that A, B, and C are consistent.
- If the component of matrix multiplication A and B, AB, is defined, then (AB) = (cA)B = A(Bc), where c is a scalar.
- Transpose: The permutation of a product of matrix multiplication A and B is (AB)T = BTAT, in which T is the transpose.
- Conjugate Complex: If A and B seem to be complex entries, then (AB)* = B*A*.
- Associativity: Multiplication of matrices is associative. Given three vectors A, B, and C that define the goods (AB)C and A(BC), then (AB)C = A (BC).
For Example: The order of Matrix A is 41. Matrix B has the 13th order. Are both the AB and BA products defined?
Solution: The order of Matrix A is 41.
The order of Matrix B is 13.
The content AB is defined by the fact that the amount of columns in A equals the same number of rows in B. Using amplification of the matrix multiplication rule, the resulting product matrix is of order 43. Because the column in B isn’t equivalent to the number of lines in A, the item BA is not defined. As a result, the multiplier is not composable, AB is not always equal to BA, and one of the commodities is not always defined.
Transpose of a Matrix is among the most frequently used composite transformation techniques in linear algebra. For a given matrix, the transpose is modified by varying the lines into columns and the columns into rows. This is especially useful for applications requiring the exact reverse and adjoint of matrices. A matrix’s transpose is obtained by reversing its rows. A matrix is a rectangular sequence of numbers or functional areas arranged in rows and columns. This variety of numbers is referred to as an entry or an element of a matrix.
Also, if you are preparing for CBSE Board exams, then these pages might surely help you:
Matrix is still a Latin word that means something is created or formed. This is a powerful and useful tool for mathematical analysis and data collection. A rectangle array of representatives with m row and column and n articles is a M x N matrix. The array’s number is referred to as an element of a matrix. The measurements of the matrix are the statistics m and n. R m x n denotes m x n matrices. Matrices are extremely important in comp sci and numerical methods.
Example 1: Which one of the preceding is not a matrix type?
a) Square matrices
b) Diagonal matrix multiplication
c) A matrix of rows
d) The minor matrix
Solution: d) A slight matrix cannot be thought of as a composite type. Matrix types include block, diagonal, and row matrices.
1. How do you identify types of Matrices?
Checking the dimension of the mixture is one way to determine its type. A matrix’s dimension is the maximum number of columns and rows in a matrix. Consider the matrix B = [1 2 5 7 0] as an example. This matrix has one row and five columns, so its aspect is 1 5. If a structure has one line and so many columns, it is referred to as a row mixture, and therefore matrix B is indeed a line matrix.
2. Which Matrix Can Never Be Inverted?
Only if the determinant of a matrix form is not zero because it is invertible. A 2 x 2 matrix, for example, is only invertible if its determinant is not 0. If a determinant of this matrix is zero, the structure is also not invertible and has no reverse. As a result, any singular matrix with a determinant equal to 0 is never invertible.
3. What Exactly Is a Triangular Matrix?
A triangle matrix is a subset of a square matrix where either all of the entries above or all of the entries underneath the diagonal elements seem to be nil. This same tridiagonal matrix and the tridiagonal matrix are the 2 kinds of triangular matrices. U m – dimensional matrix is a matrix in which all elements below the diagonal matrix are 0. A lower triangular square matrix in which all the elements above the diagonal are 0.
4. A 2×2 matrix is what type of matrix?
Square matrix is one with dimension n n, which means it has an equivalent number of rows. A 2 2 matrix has two rows and two columns, making it a square matrix.