{"id":28194,"date":"2022-01-17T19:31:05","date_gmt":"2022-01-17T14:01:05","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=28194"},"modified":"2022-01-17T19:31:05","modified_gmt":"2022-01-17T14:01:05","slug":"matrices-class-12-notes-maths-chapter-3","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-matrices\/matrices\/class-12-notes\/maths-chapter-3\/","title":{"rendered":"Matrices Class 12 Notes Maths Chapter 3"},"content":{"rendered":"

Matrix:<\/strong> A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.<\/p>\n

Order of a Matrix:<\/strong> If a matrix has m rows and n columns, then its order is written as m \u00d7 n. If a matrix has order m \u00d7 n, then it has mn elements.<\/p>\n

In general, am\u00d7n<\/sub> matrix has the following rectangular array:
\n\"Matrices
\nNote: We shall consider only those matrices, whose elements are real numbers or functions taking real values.<\/p>\n

Types of Matrices<\/strong>
\nColumn Matrix:<\/strong> A matrix which has only one column, is called a column matrix.
\ne.g. \\(\\left[ \\begin{matrix} 1 \\\\ 0 \\\\ -5 \\end{matrix} \\right]\\)
\nIn general, A = [aij<\/sub>]m\u00d71<\/sub> is a column matrix of order m \u00d7 1.<\/p>\n

Row Matrix:<\/strong> A matrix which has only one row, is called a row matrix,
\ne.g. \\(\\left[ \\begin{matrix} 1 & 5 & 9 \\end{matrix} \\right]\\)
\nIn general, A = [aij<\/sub>]1\u00d7n<\/sub> is a row matrix of order 1 x n<\/p>\n

Square Matrix:<\/strong> A matrix which has equal number of rows and columns, is called a square matrix
\ne.g. \\(\\begin{bmatrix} 3 & -1 \\\\ 5 & 2 \\end{bmatrix}\\)
\nIn general, A = [aij<\/sub>]m x m is a square matrix of order m.
\nNote: If A = [aij<\/sub>] is a square matrix of order n, then elements a11<\/sub>, a22<\/sub>, a33<\/sub>,\u2026, ann <\/sub>is said to constitute the diagonal of the matrix A.<\/p>\n

Diagonal Matrix:<\/strong> A square matrix whose all the elements except the diagonal elements are zeroes, is called a diagonal matrix,
\ne.g. \\(\\left[ \\begin{matrix} 3 & 0 & 0 \\\\ 0 & -3 & 0 \\\\ 0 & 0 & -8 \\end{matrix} \\right]\\)
\nIn general, A = [aij<\/sub>]m\u00d7m<\/sub> is a diagonal matrix, if aij<\/sub> = 0, when i \u2260 j.<\/p>\n

Scalar Matrix:<\/strong> A diagonal matrix whose all diagonal elements are same (non-zero), is called a scalar matrix,
\ne.g. \\(\\left[ \\begin{matrix} 2 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\end{matrix} \\right]\\)
\nIn general, A = [aij<\/sub>]n\u00d7n<\/sub> is a scalar matrix, if aij<\/sub> = 0, when i \u2260 j, aij<\/sub> = k (constant), when i = j.
\nNote: A scalar matrix is a diagonal matrix but a diagonal matrix may or may not be a scalar matrix.<\/p>\n

Unit or Identity Matrix:<\/strong> A diagonal matrix in which all diagonal elements are \u20181\u2019 and all non-diagonal elements are zero, is called an identity matrix. It is denoted by I.
\ne.g. \\(\\left[ \\begin{matrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{matrix} \\right]\\)
\nIn general, A = [aij<\/sub>]n\u00d7n<\/sub> is an identity matrix, if aij<\/sub> = 1, when i = j and aij<\/sub> = 0, when i \u2260 j.<\/p>\n

Zero or Null Matrix:<\/strong> A matrix is said to be a zero or null matrix, if its all elements are zer0
\ne.g. \\(\\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix}\\)<\/p>\n

Equality of Matrices:<\/strong> Two matrices A and B are said to be equal, if
\n(i) order of A and B are same.
\n(ii) corresponding elements of A and B are same i.e. aij<\/sub> = bij<\/sub>, \u2200 i and j.
\ne.g. \\(\\begin{bmatrix} 2 & 1 \\\\ 0 & 3 \\end{bmatrix}\\) and \\(\\begin{bmatrix} 2 & 1 \\\\ 0 & 3 \\end{bmatrix}\\) are equal matrices, but \\(\\begin{bmatrix} 3 & 2 \\\\ 0 & 1 \\end{bmatrix}\\) and \\(\\begin{bmatrix} 2 & 3 \\\\ 0 & 1 \\end{bmatrix}\\) are not equal matrices.<\/p>\n

Operations on Matrices<\/strong>
\nBetween two or more than two matrices, the following operations are defined below:
\nAddition and Subtraction of Matrices:<\/strong> Addition and subtraction of two matrices are defined in an order of both the matrices are same.
\nAddition of Matrix
\nIf A = [aij<\/sub>]m\u00d7n<\/sub> and B = [yij<\/sub>]m\u00d7n<\/sub>, then A + B = [aij<\/sub> +bij<\/sub>]m\u00d7n<\/sub>, 1 \u2264 i \u2264 m, 1 \u2264 j \u2264 n
\nSubtraction of Matrix
\nIf A = [aij<\/sub>]m\u00d7n<\/sub> and B = [bij<\/sub>]m\u00d7n<\/sub>, then A \u2013 B = [aij<\/sub> \u2013 bij<\/sub>]m\u00d7n<\/sub>, 1 \u2264 i \u2264 m, 1 \u2264 j \u2264 n<\/p>\n

Properties of Addition of Matrices<\/strong>
\n(a) Commutative If A = [aij<\/sub>] and B = [bij<\/sub>] are matrices of the same order say m x n then A + B = B + A,
\n(b) Associative for any three matrices A = [aij<\/sub>], B = [bij<\/sub>], C = [cij<\/sub>] of the same order say m x n, A + (B + C) = (A + B) + C.
\n(c) Existence of additive identity Let A = [aij] be amxn matrix and O be amxn zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition.
\n(d) Existence of additive inverse Let A = [aij<\/sub>]m\u00d7n<\/sub> be any matrix, then we have another matrix as -A = [-aij<\/sub>]m\u00d7n<\/sub> such that A + (-A) = (-A + A) = O. So, matrix (-A) is called additive inverse of A or negative of A.<\/p>\n

Note
\n(i) If A and B are not of the same order, then A + B is not defined.
\n(ii) Addition of matrices is an example of a binary operation on the set of matrices of the same order.<\/p>\n

Multiplication of a matrix by scalar number:<\/strong> Let A = [aij<\/sub>]m\u00d7n<\/sub> be a matrix and k is scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e. if A = [aij<\/sub>]m\u00d7n<\/sub>, then kA = [kaij<\/sub>]m\u00d7n<\/sub>.
\n\"Matrices<\/p>\n

Properties of Scalar Multiplication of a Matrix
\nLet A = [aij<\/sub>] and B = [bij<\/sub>]be two matrices of the same order say m \u00d7 n, then
\n(a) k(A + B) = kA + kB, where k is a scalar.
\n(b) (k + l)A = kA + lA, where k and l are scalars.<\/p>\n

Multiplication of Matrices:<\/strong> Let A and B be two matrices. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B.
\n\"Matrices<\/p>\n

Properties of Multiplication of Matrices
\n(a) Non-commutativity Matrix multiplication is not commutative i.e. if AB and BA are both defined, then it is not necessary that AB \u2260 BA.
\n(b) Associative law For three matrices A, B, and C, if multiplication is defined, then A (BC) = (AB) C.
\n(c) Multiplicative identity For every square matrix A, there exists an identity matrix of the same order such that IA = AI = A.
\nNote: For Amxm, there is only one multiplicative identity Im<\/sub>.
\n(d) Distributive law For three matrices A, B, and C,
\nA(B + C) = AB + AC
\n(A + B)C = AC + BC
\nwhenever both sides of the equality are defined.<\/p>\n

Note: If A and B are two non-zero matrices, then their product may be a zero matrix.
\ne.g. Suppose A = \\(\\begin{bmatrix} 0 & -1 \\\\ 0 & 2 \\end{bmatrix}\\) and B = \\(\\begin{bmatrix} 3 & 5 \\\\ 0 & 0 \\end{bmatrix}\\), then AB = \\(\\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix}\\).<\/p>\n

<\/h5>\n","protected":false},"excerpt":{"rendered":"

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