{"id":64880,"date":"2025-06-20T09:43:31","date_gmt":"2025-06-20T06:43:31","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64880"},"modified":"2026-09-30T20:19:29","modified_gmt":"2026-09-30T17:19:29","slug":"matrix-addition","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/matrix-addition\/","title":{"rendered":"Addition of Two Matrices Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Matrix Addition Solver<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"64880\" data-favorite-title=\"Addition of Two Matrices\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-addition\/\" data-favorite-excerpt=\"Add two matrices of the same size (2\u00d72 or 3\u00d73) by entering their elements. The calculator sums corresponding entries and displays the resulting matrix instantly.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-addition\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64880\" data-post=\"64880\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">\u0420\u0430\u0437\u043c\u0435\u0440 \u043c\u0430\u0442\u0440\u0438\u0446<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"razmer_matryts\"><option value=\"2\">2 \u00d7 2<\/option><option value=\"3\">3 \u00d7 3<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"is-single field\" id=\"input-2\"><span class=\"font-semibold\">\u041c\u0430\u0442\u0440\u0438\u0446\u0430 A<\/span><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" 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aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused 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border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Matrix Addition Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ C = A + B = \\begin{bmatrix} a_{11} + b_{11} &amp; a_{12} + b_{12} &amp; \\cdots &amp; a_{1n} + b_{1n} \\\\ a_{21} + b_{21} &amp; a_{22} + b_{22} &amp; \\cdots &amp; a_{2n} + b_{2n} \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ a_{m1} + b_{m1} &amp; a_{m2} + b_{m2} &amp; \\cdots &amp; a_{mn} + b_{mn} \\end{bmatrix} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(A\\) and \\(B\\) \u2014 matrices of identical size \\(m \\times n\\)<\/li>\n<li>\\(C\\) \u2014 the resulting matrix of the same size \\(m \\times n\\)<\/li>\n<li>\\(a_{ij}\\), \\(b_{ij}\\) \u2014 the elements in row \\(i\\), column \\(j\\) of matrices \\(A\\) and \\(B\\)<\/li>\n<\/ul>\n<p>Each element of the result is the sum of the corresponding elements of the two matrices: \\(c_{ij} = a_{ij} + b_{ij}\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Addition of Two Matrices \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> add a 3\u00d73 matrix \\(A\\) to the identity matrix \\(B\\), using the calculator's default values.<\/p>\n<p>\\[ A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 \\end{bmatrix}, \\quad B = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p>\\[ C = A + B = \\begin{bmatrix} 1+1 &amp; 2+0 &amp; 3+0 \\\\ 4+0 &amp; 5+1 &amp; 6+0 \\\\ 7+0 &amp; 8+0 &amp; 9+1 \\end{bmatrix} = \\begin{bmatrix} 2 &amp; 2 &amp; 3 \\\\ 4 &amp; 6 &amp; 6 \\\\ 7 &amp; 8 &amp; 10 \\end{bmatrix} \\]<\/p>\n<p><strong>Result:<\/strong> the sum of the two matrices is \\(C = \\begin{bmatrix} 2 &amp; 2 &amp; 3 \\\\ 4 &amp; 6 &amp; 6 \\\\ 7 &amp; 8 &amp; 10 \\end{bmatrix}\\). Notice how adding the identity matrix leaves every element of \\(A\\) unchanged except the diagonal entries, which each increase by 1.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Choose the <strong>Matrix size<\/strong> \u2014 either 2 \u00d7 2 or 3 \u00d7 3 \u2014 from the dropdown at the top. Both matrices must have the same size.<\/li>\n<li>Enter the elements of <strong>Matrix A<\/strong> in the fields \\(a_{11}, a_{12}, \\ldots\\). Each subscript shows the row and column of the element (for example, \\(a_{21}\\) is row 2, column 1).<\/li>\n<li>Enter the elements of <strong>Matrix B<\/strong> in the same way, using the fields \\(b_{11}, b_{12}, \\ldots\\).<\/li>\n<li>The result appears immediately: a table showing the matrix \\(C = A + B\\) with the sums of corresponding elements. Values are rounded to two decimal places.<\/li>\n<\/ol>\n<h2>What is matrix addition?<\/h2>\n<p>Matrix addition is one of the simplest operations in linear algebra. To add two matrices, you add their corresponding elements \u2014 the entry in row \\(i\\), column \\(j\\) of the result equals the sum of the entries in the same position of the two original matrices. The operation is defined only for matrices of identical dimensions: a 2\u00d72 matrix can be added to a 2\u00d72 matrix, but not to a 3\u00d73 one, because there would be elements with no counterpart to add.<\/p>\n<p>Unlike matrix multiplication, which is order-dependent and far more involved, addition works element by element and behaves much like ordinary addition of numbers. That makes it a natural starting point when learning matrix operations, and it is used constantly in applied work: summing forces or displacements in physics and engineering, combining pixel data in image processing, accumulating weights in machine learning, and aggregating tables of values in economics and statistics.<\/p>\n<p>The interpretation of the result is straightforward: each element of \\(C\\) describes the combined effect of the corresponding elements of \\(A\\) and \\(B\\). For example, if two matrices record the monthly sales of different products in two stores, their sum gives the combined sales across both stores. Adding the identity matrix, as in the example above, leaves all off-diagonal elements unchanged and increases each diagonal element by 1.<\/p>\n<h2>Properties of matrix addition<\/h2>\n<p>Matrix addition satisfies the same familiar rules as addition of real numbers:<\/p>\n<ul>\n<li><strong>Commutativity:<\/strong> \\(A + B = B + A\\) \u2014 the order of the matrices does not matter.<\/li>\n<li><strong>Associativity:<\/strong> \\((A + B) + C = A + (B + C)\\) \u2014 grouping does not affect the result.<\/li>\n<li><strong>Identity element:<\/strong> \\(A + O = A\\), where \\(O\\) is the zero matrix of the same size.<\/li>\n<li><strong>Inverse element:<\/strong> \\(A + (-A) = O\\), where \\(-A\\) has every element of \\(A\\) with the opposite sign.<\/li>\n<\/ul>\n<p>Keep in mind that these properties apply to addition, not to multiplication: in general \\(AB \\neq BA\\), and \\(A + B\\) has no direct analogue in multiplication. If you need the product of two matrices, use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-multiplication\/\">Matrix Multiplication<\/a> calculator; for the difference of two matrices, see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-subtraction\/\">Subtraction of Two Matrices<\/a> calculator.<\/p>\n<h2>Assumptions and limitations<\/h2>\n<ul>\n<li><strong>Equal dimensions required.<\/strong> The calculator only adds matrices of the same size. Matrices of different dimensions cannot be added by definition.<\/li>\n<li><strong>Supported sizes.<\/strong> Only 2 \u00d7 2 and 3 \u00d7 3 matrices are supported. For larger matrices, the same element-by-element rule applies, but you would need a more general tool.<\/li>\n<li><strong>Numeric entries.<\/strong> All elements must be numbers (integers or decimals). Fractional values should be converted to decimal form before entry.<\/li>\n<li><strong>Rounding.<\/strong> Results are displayed rounded to two decimal places, which is sufficient for most coursework and practical applications.<\/li>\n<li><strong>Determinants do not add.<\/strong> In general, \\(\\det(A + B) \\neq \\det(A) + \\det(B)\\). If you need the determinant of the resulting matrix or want to compare with <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-of-determinants\/\">Sum of Determinants<\/a>, compute it from the final matrix, not from the individual ones.<\/li>\n<\/ul>\n<h2>Practical tips<\/h2>\n<ul>\n<li>Double-check that you have entered elements in the correct positions \u2014 row first, then column. A common mistake is transposing an element, e.g. entering \\(a_{21}\\) in the \\(a_{12}\\) field.<\/li>\n<li>Use the result to verify hand calculations: compute one or two elements yourself and compare them with the calculator's output.<\/li>\n<li>For related operations, you may also find the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/inverse-matrix-3x3\/\">Inverse Matrix 3\u00d73<\/a> and <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/rank-3x3-matrix\/\">Rank of 3\u00d73 Matrix<\/a> calculators useful when analyzing the resulting matrix further.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Add two matrices of the same size (2\u00d72 or 3\u00d73) by entering their elements. The calculator sums corresponding entries and displays the resulting matrix instantly.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[69],"class_list":["post-64880","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-matrix"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64880","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=64880"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64880\/revisions"}],"predecessor-version":[{"id":66202,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64880\/revisions\/66202"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64880"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64880"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64880"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}