{"id":35029,"date":"2022-08-25T09:54:54","date_gmt":"2022-08-25T04:24:54","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35029"},"modified":"2022-08-25T09:54:54","modified_gmt":"2022-08-25T04:24:54","slug":"inter-1st-year-maths-1a-matrices-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-1st-year-maths-1a-matrices-formulas\/","title":{"rendered":"Inter 1st Year Maths 1A Matrices Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.guru\/inter-1st-year-maths-1a-formulas\/\">Inter 1st Year Maths 1A Formulas<\/a> PDF Chapter 3 Matrices to solve questions creatively.<\/p>\n<h2>Intermediate 1st Year Maths 1A Matrices Formulas<\/h2>\n<p>\u2192 An ordered rectangular array of elements is called a matrix.<\/p>\n<p>\u2192 A matrix in which the number of rows is equal to the number of columns, is called a square matrix. Otherwise, it is called a rectangular matrix. In a square matrix, an element a<sub>ij<\/sub> is in principal diagonal, if i = j<\/p>\n<p>\u2192 If each non-diagonal element of a square matrix is equal to zero, then it is called a diagonal matrix.<\/p>\n<p>\u2192 If each non-diagonal element of a square matrix is equal to zero and each diagonal element is equal to a scalar, then it is called a scalar matrix.<\/p>\n<p>\u2192 If each non-diagonal element of a square matrix is equal to zero and each diagonal element is equal to 1, then that matrix is called a unit matrix or Identity matrix.<\/p>\n<p>\u2192 If A is a square matrix then the sum of elements in the principal diagonal of A is called trace of A.<\/p>\n<ul>\n<li>Matrix addition is commutative.<\/li>\n<li>Matrix addition is associative.<\/li>\n<li>Matrix multiplication is associative.<\/li>\n<li>Matrix multiplication is distributive over matrix addition.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 A square matrix A is said to be an idempotent matrix if A<sup>2<\/sup> = A.<\/p>\n<p>\u2192 A square matrix A is said to be an involuntary matrix if A<sup>2<\/sup> = I.<\/p>\n<p>\u2192 A square matrix A is said to be a nilpotent matrix, if there exist a positive integer n such that A<sup>n<\/sup> = 0. If n is the least positive integer such thatAn = 0, then n is called index of the nilpotent matrix A.<\/p>\n<p>\u2192 The matrix obtained by interchanging rows and columns is called transpose of the given matrix. Transpose of A is denoted by A<sup>T<\/sup> (or) A&#8217;.<\/p>\n<p>\u2192 For any matrices A and B<\/p>\n<ul>\n<li>(A<sup>T<\/sup>)<sup>T<\/sup> = A<\/li>\n<li>(A + B)<sup>T<\/sup> = A<sup>T<\/sup> + B<sup>T<\/sup> (A and B are same order)<\/li>\n<li>(AB)<sup>T<\/sup> = B<sup>T<\/sup>A<sup>T<\/sup>. (If A and B are of orders m xn and n xp respectively)<\/li>\n<\/ul>\n<p>\u2192 A square matrix A is said to be symmetric if A<sup>T<\/sup> = A.<\/p>\n<p>\u2192 A square matrix A is said to be skew symmetric if A<sup>T<\/sup> = A.<\/p>\n<p>\u2192 Every square matrix can be uniquely expressed as a sum of symmetric matrix and a skew symmetric matrix.<\/p>\n<p>\u2192 The minor of an element in the square matrix of order &#8216;3&#8217; is defined as the determinant of 2 \u00d7 2 matrix, obtained after deleting the row and the column in which the element is present.<\/p>\n<p>\u2192 The cofactor of an element in the i<sup>th<\/sup> row and j<sup>th<\/sup> column of 3 \u00d7 3 matrix is defined as its minor multiplied by (-1)<sup>i+j<\/sup>.<\/p>\n<p>\u2192 The sum of the products of the elements of any row or column with their corresponding cofactors is called the determinant of a matrix.<\/p>\n<ul>\n<li>A square &#8216;A&#8217; is said to be a singular matrix if det A = 0.<\/li>\n<li>A square A&#8217; is said to be a non-singular matrix if det A \u2260 0.<\/li>\n<\/ul>\n<p>\u2192 The transpose of the matrix obtained by replacing.the elements of a square matrix A by the corresponding cofactors is called the adjoint matrix of A and it is denoted by adj (A).<\/p>\n<ul>\n<li>A square matrix &#8216;A is said to be an invertible matrix if there exists a square matrix B such that AB = BA = I the matrix B is called inverse of A and it is denoted by A<sup>-1<\/sup>.<\/li>\n<li>If A is an invertible then A<sup>-1<\/sup> = 1.<\/li>\n<\/ul>\n<p>\u2192 A square matrix A is non-singular if A is invertible.<\/p>\n<ul>\n<li>If A and B are non-singular matrices of same type then Adj (AB) = (Adj B) (Adj A).<\/li>\n<li>If A is a square matrix of type n then det (Adj A) = (det A)<sup>n-1<\/sup>.<\/li>\n<\/ul>\n<p>\u2192 A matrix obtained by deleting some rows or columns (or both) of a matrix is called a submatrix.<\/p>\n<p>\u2192 Let Abe a non-zero matrix, the rank of A is defined as the maximum of the orders of the non-singular square submatrices of A. The rank of a null matrix is zero. The rank of a matrix A is denoted as rank (A) or P (A).<\/p>\n<p>\u2192 A system of linear equations is<\/p>\n<ul>\n<li>Consistent, if it has a solution<\/li>\n<li>Inconsistent, if it has no solution<\/li>\n<\/ul>\n<p>\u2192 Non homogeneous system<\/p>\n<ul>\n<li>a<sub>1<\/sub>x + b<sub>1<\/sub>y + c<sub>1<\/sub>z = d<sub>1<\/sub><\/li>\n<li>a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub>z = d<sub>2<\/sub><\/li>\n<li>a<sub>3<\/sub>x + b<sub>3<\/sub>y + c<sub>3<\/sub>z = d<sub>3<\/sub><\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 The above system of equations has<\/p>\n<ul>\n<li>aunique solution if rank (A) = Rank [AD] = 3<\/li>\n<li>infinity many solutions, if rank (A) = Rank ([AD]) &lt; 3<\/li>\n<li>no solution, if rank A \u2260 Rank ([AD])<\/li>\n<\/ul>\n<p>\u2192 Homogeneous system of equations<\/p>\n<ul>\n<li>a<sub>1<\/sub>x + b<sub>1<\/sub>y + c<sub>1<\/sub>z = d<sub>1<\/sub><\/li>\n<li>a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub>z = d<sub>2<\/sub><\/li>\n<li>a<sub>3<\/sub>x + b<sub>3<\/sub>y + c<sub>3<\/sub>z = d<sub>3<\/sub><\/li>\n<\/ul>\n<p>\u2192 The above system has<\/p>\n<ul>\n<li>Trival solution x = y = z = 0 only if rank (A) = 3<\/li>\n<li>infinitely many non-trival solutions if rank (A) &lt; 3.<\/li>\n<\/ul>\n<p>\u2192 A matrix is an arrangement of real or complex numbers into rows and columns so that all the rows (columns) contain equal no. of elements.<\/p>\n<p>\u2192 If a matrix consists of \u2018m\u2019 rows and \u2018n\u2019 columns, then it is said to be of order m \u00d7 n.<\/p>\n<p>\u2192 A matrix of order n \u00d7 n is said to be a square matrix of order n.<\/p>\n<p>\u2192 A matrix (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub> is said to be a null matrix if a<sub>ij<\/sub> = 0 for all i and j.<\/p>\n<p>\u2192 Two matrices of the same order are said to be equal if the corresponding elements in the matrices are all equal.<\/p>\n<p>\u2192 A matrix (a<sub>ij<\/sub>)<sub>n\u00d7n<\/sub> is a diagonal matrix a<sub>ij<\/sub> = 0 for all i \u2260 j<\/p>\n<p>\u2192 A matrix (a<sub>ij<\/sub>)<sub>n\u00d7n<\/sub> is a scalar matrix if a = 0 for all i \u2260 j and a<sub>ij<\/sub> = k (constant) for i = j<\/p>\n<p>\u2192 A matrix (a<sub>ij<\/sub>)<sub>n\u00d7n<\/sub> is said to be a unit matrix of order n, denoted by I<sub>n<\/sub> if a<sub>ij<\/sub> = 1, when i = j and a<sub>ij<\/sub> = 0 when i \u2260 j<br \/>\nEx: I<sub>2<\/sub> = \\(\\left[\\begin{array}{ll}<br \/>\n1 &amp; 0 \\\\<br \/>\n0 &amp; 1<br \/>\n\\end{array}\\right]\\)<br \/>\nI<sub>3<\/sub> = \\(\\left[\\begin{array}{lll}<br \/>\n1 &amp; 0 &amp; 0 \\\\<br \/>\n0 &amp; 1 &amp; 0 \\\\<br \/>\n0 &amp; 0 &amp; 1<br \/>\n\\end{array}\\right]\\)<\/p>\n<p>\u2192 If A = (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>, B = (b<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>, then A + B = (a<sub>ij<\/sub> + b<sub>ij<\/sub>)<sub>m\u00d7n<\/sub><\/p>\n<p>\u2192 Matrix addition is commutative and associative<\/p>\n<p>\u2192 Matrix multiplication is not commutative but associative<\/p>\n<p>\u2192 If A is a matrix of order m \u00d7 n, then AI<sub>n<\/sub> = I<sub>m<\/sub>A = A(AI = IA = A)<\/p>\n<p>\u2192 If AB = CA = I, then B = C<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 If A = (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>, then A <sup>T<\/sup> = (a<sub>ij<\/sub>)<sub>n\u00d7m<\/sub><\/p>\n<p>\u2192 (KA)<sup>T<\/sup> = KA<sup>T<\/sup>, (A + B)<sup>T<\/sup> = A<sup>T<\/sup> + B<sup>T<\/sup>, (AB)<sup>T<\/sup> = B<sup>T<\/sup>.A<sup>T<\/sup><\/p>\n<p>\u2192 A(B + C) = AB + AC, (A + B)C = AC + BC<\/p>\n<p>\u2192 A square matrix is said to be \u201cnon-singular\u201d if detA \u2260 0<\/p>\n<p>\u2192 A square matrix is said to be \u201csingular\u201d if detA = 0<\/p>\n<p>\u2192 If AB = 0, where A and B are non-zero square matrices, then both A are singular.<\/p>\n<p>\u2192 A minor of any element in a square matrix is determinant of the matrix obtained by omitting the row and column in which the element is present.<\/p>\n<p>\u2192 In (a<sub>ij<\/sub>)<sub>n\u00d7n<\/sub>, the cofactor of a<sub>ij<\/sub> is (-1)<sup>i+j<\/sup> \u00d7 (minor of a<sub>ij<\/sub>).<\/p>\n<p>\u2192 In a square matrix, the sum of the products of the elements of any row (column) and the corresponding cofactors is equal to the determinant of the matrix.<\/p>\n<p>\u2192 In a square matrix, the sum of the products of the elements of any row (column) and the corresponding cofactors of any other row (column) is alway s zero.<\/p>\n<p>\u2192 If A is any square matrix, then A adjA = adjA. A = detA. I<\/p>\n<p>\u2192 If A is any square matrix and there exists a matrix B such that AB = BA = I, then B is called the inverse of A and denoted by A<sup>-1<\/sup>.<\/p>\n<p>\u2192 AA<sup>-1<\/sup> = A<sup>-1<\/sup>A = I.<\/p>\n<p>\u2192 If A is non-singular, then A<sup>-1<\/sup> = \\(\\frac{{adj} A}{{det} A}\\) (or) adj A = |A|AA<sup>-1<\/sup><\/p>\n<p>\u2192 If A = \\(\\left(\\begin{array}{ll}<br \/>\na &amp; b \\\\<br \/>\nc &amp; d<br \/>\n\\end{array}\\right)\\), then A<sup>-1<\/sup> = \\(\\frac{1}{a d-b c}\\left(\\begin{array}{cc}<br \/>\nd &amp; -b \\\\<br \/>\n-c &amp; a<br \/>\n\\end{array}\\right)\\)<\/p>\n<p>\u2192 (A<sup>-1<\/sup>)<sup>-1<\/sup> = A, (AB)<sup>-1<\/sup> = B<sup>-1<\/sup>.A<sup>-1<\/sup>, (A<sup>-1<\/sup>)<sup>T<\/sup> =( A<sup>T<\/sup>)<sup>-1<\/sup>; (ABC&#8230;.)<sup>-1<\/sup> = C<sup>-1<\/sup>B<sup>-1<\/sup>A<sup>-1<\/sup>.<\/p>\n<p>Theorem:<br \/>\nMatrix multiplication is associative. i.e. if conformability is assured for the matrices A, B and C, then (AB)C = A(BC).<br \/>\nProof:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-19652\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/Inter-1st-Year-Maths-1A-Matrices-Formulas-1.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas 1\" width=\"335\" height=\"556\" \/><\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Theorem:<br \/>\nMatrix multiplication is distributive over matrix addition i.e. if conformability is assured for the matrices A, B and C, then<br \/>\n(i) A (B + C) = AB + AC<br \/>\n(ii) (B + C) A = BA + CA<br \/>\nProof:<br \/>\nLet A = (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>, B = (b<sub>jk<\/sub>)<sub>n\u00d7p<\/sub> C = (c<sub>ki<\/sub>)<sub>n\u00d7p<\/sub><br \/>\nB + C = (d<sub>jk<\/sub>)<sub>n\u00d7p<\/sub>, where d<sub>jk<\/sub> = b<sub>jk<\/sub> + c<sub>jk<\/sub><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-19651\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/Inter-1st-Year-Maths-1A-Matrices-Formulas-2.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas 2\" width=\"335\" height=\"396\" \/><br \/>\n\u2234 A(B + C) = AB + AC<br \/>\nSimilarly we can prove that<br \/>\n(B + C) = BA + CA.<\/p>\n<p>Theorem:<br \/>\nIf A is any matrix, then (A<sup>T<\/sup>)<sup>T<\/sup> = A.<br \/>\nProof:<br \/>\nLet A = (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub><br \/>\nA<sup>T<\/sup> = (a&#8217;<sub>jk<\/sub>)<sub>n\u00d7m<\/sub>, where a&#8217;<sub>ji<\/sub> = a<sub>ij<\/sub><br \/>\n(A<sup>T<\/sup>)<sup>T<\/sup> = (a&#8221;<sub>ji<\/sub>)<sub>m\u00d7n<\/sub>, where a&#8221;<sub>ij<\/sub> = a<sub>ji<\/sub><br \/>\na&#8221;<sub>ij<\/sub> = a&#8217;<sub>ji<\/sub> = a<sub>ij<\/sub><br \/>\n\u2234 (A<sup>T<\/sup>)<sup>T<\/sup> = A<\/p>\n<p>Theorem:<br \/>\nIf A and B are two matrices o same type, then (A + B)<sup>T<\/sup> = A<sup>T<\/sup> + B<sup>T<\/sup>.<br \/>\nProof:<br \/>\nLet A = (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>, B = (b<sub>ij<\/sub>)<br \/>\nA + B = (c<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>,where c<sub>ij<\/sub> = a<sub>ij<\/sub> + b<sub>ij<\/sub><br \/>\n(A + B)<sup>T<\/sup> = (c&#8217;<sub>ji<\/sub>)<sub>n\u00d7m<\/sub>. c&#8217;<sub>ji<\/sub> = c<sub>ij<\/sub><br \/>\nA<sup>T<\/sup> = (a&#8217;<sub>ji<\/sub>)<sub>n\u00d7m<\/sub>,where a&#8217;<sub>ji<\/sub> = a<sub>ij<\/sub><br \/>\nB<sup>T<\/sup> = (b<sub>ji<\/sub>)<sub>n\u00d7m<\/sub>. where. b&#8217;<sub>kj<\/sub> = b<sub>jk<\/sub><br \/>\nA<sup>T<\/sup> + B<sup>T<\/sup> = (d<sub>ji<\/sub>)<sub>n\u00d7m<\/sub>, where d<sub>ji<\/sub> = a&#8217;<sub>ji<\/sub> + b&#8217;<sub>ji<\/sub><br \/>\nc&#8217;<sub>ji<\/sub> = c<sub>ij<\/sub> = a<sub>ij<\/sub> + b<sub>ij<\/sub> = a&#8217;<sub>ji<\/sub> + b&#8217;<sub>ji<\/sub> = d<sub>ji<\/sub><br \/>\n\u2234(A + B)<sup>T<\/sup> = A<sup>T<\/sup> + B<sup>T<\/sup><\/p>\n<p>Theorem:<br \/>\nIf A and B are two matrices for which conformability for multiplication is assured, then (AB)<sup>T<\/sup> = B<sup>T<\/sup>A<sup>T<\/sup>.<br \/>\nPr0of:<br \/>\nLet A = (a<sub>ij<\/sub>)<sub>m\u00d7n<\/sub>, B = (b<sub>ji<\/sub>)<sub>n\u00d7p<\/sub><br \/>\nAB = (c<sub>ik<\/sub>)<sub>m\u00d7p<\/sub>, where c<sub>ik<\/sub> = \\(\\sum_{j=1}^{n}\\) a<sub>ij<\/sub>b<sub>jk<\/sub><br \/>\n(AB)<sup>T<\/sup> = (c<sub>ki<\/sub>)<sub>p\u00d7m<\/sub>,where c<sub>ki<\/sub> = c<sub>ik<\/sub><br \/>\nA<sup>T<\/sup> = (a<sub>ji<\/sub>)<sub>n\u00d7m<\/sub>,where a<sub>ji<\/sub> = a<sub>ij<\/sub><br \/>\nB<sup>T<\/sup> = (b<sub>kj<\/sub>)<sub>p\u00d7n<\/sub>, where b<sub>kj<\/sub> = b<sub>jk<\/sub><br \/>\nB<sup>T<\/sup> . A<sup>T<\/sup> = (d<sub>ki<\/sub> )<sub>p\u00d7m<\/sub>, where d<sub>ki<\/sub>= \\(\\sum_{j=1}^{n}\\) b<sub>kj<\/sub>a<sub>ji<\/sub><br \/>\nc&#8217;ki = cik = \\(\\sum_{j=1}^{n}\\) a<sub>ij<\/sub>b<sub>jk<\/sub>= \\(\\sum_{j=1}^{n}\\) b<sub>kj<\/sub>a<sub>ji<\/sub>d<sub>ki<\/sub><br \/>\n\u2234 (AB)<sup>T<\/sup> = B<sup>T<\/sup>A<sup>T<\/sup><\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>Theorem:<br \/>\nIf A and B are two invertible matrices of same type then AB is also invertible and (AB)<sup>-1<\/sup> = B<sup>-1<\/sup>A<sup>-1<\/sup>.<br \/>\nProof:<br \/>\nA is invertible matrix \u21d2 A<sup>-1<\/sup> exists and AA<sup>-1<\/sup> = A<sup>-1<\/sup>A = I.<br \/>\nB is an invertible matrix \u21d2 B<sup>-1<\/sup> exists and<br \/>\nBB<sup>-1<\/sup> = B<sup>-1<\/sup>B = I<br \/>\nNow (AB)(B<sup>-1<\/sup>A<sup>-1<\/sup>) = A(BB<sup>-1<\/sup>)A<sup>-1<\/sup> = AIA<sup>-1<\/sup> = AA<sup>-1<\/sup> = I<br \/>\n(B<sup>-1<\/sup>A<sup>-1<\/sup>)(AB) = B<sup>-1<\/sup>(A<sup>-1<\/sup>A)B \u2234 AB is invertible and<br \/>\n= B<sup>-1<\/sup>IB = B<sup>-1<\/sup>B = I<br \/>\n(AB)(B<sup>-1<\/sup>A<sup>-1<\/sup>) = (B<sup>-1<\/sup>A<sup>-1<\/sup>) = (B<sup>-1<\/sup>A<sup>-1<\/sup>)(AB) = 1<br \/>\n(AB)<sup>-1<\/sup> = B<sup>-1<\/sup>A<sup>-1<\/sup>.<\/p>\n<p>Theorem:<br \/>\nIf A is a non-singular matrix then A is invertible and A<sup>-1<\/sup> = \\(\\frac{{Adj} A}{{det} A}\\).<br \/>\nProof:<br \/>\nLet A = \\(\\left[\\begin{array}{lll}<br \/>\n\\mathrm{a}_{1} &amp; \\mathrm{~b}_{1} &amp; \\mathrm{c}_{1} \\\\<br \/>\n\\mathrm{a}_{2} &amp; \\mathrm{~b}_{2} &amp; \\mathrm{c}_{2} \\\\<br \/>\n\\mathrm{a}_{3} &amp; \\mathrm{~b}_{3} &amp; \\mathrm{c}_{3}<br \/>\n\\end{array}\\right]\\) be a non &#8211; singular matrix.<br \/>\n\u2234 det A \u2260 0<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-19650\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/Inter-1st-Year-Maths-1A-Matrices-Formulas-3.png\" alt=\"Inter 1st Year Maths 1A Matrices Formulas 3\" width=\"478\" height=\"715\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 1st Year Maths 1A Formulas PDF Chapter 3 Matrices to solve questions creatively. Intermediate 1st Year Maths 1A Matrices Formulas \u2192 An ordered rectangular array of elements is called a matrix. \u2192 A matrix in which the number of rows is equal to the number of columns, is called a square matrix. &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[15],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35029"}],"collection":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/comments?post=35029"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35029\/revisions"}],"predecessor-version":[{"id":35030,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35029\/revisions\/35030"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/media?parent=35029"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/categories?post=35029"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/tags?post=35029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}