neve domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home/mathze5/public_html/learnermath.com/wp-includes/functions.php on line 6114Matrix multiplication becomes a bit more involved when we look at multiplying matrices examples where matrices are multiplied together.\n We just multiply each element of the matrix by the number.\n We can look at the case of multiplying a 2 x 2<\/font> matrix by another 2 x 2 <\/font> matrix.<\/p>\n\\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix}<\/span> × \\begin{bmatrix} b_{11} & b_{12} \\\\ b_{21} & b_{22} \\end{bmatrix}<\/span>\n The result of multiplying two 2 x 2<\/font> matrices together was another 2 x 2<\/font> matrix. \n It’s important to pay attention to order when learning matrix multiplication.\n A specific case to consider when learning how to do matrix multiplication, is that multiplication of a matrix with the identity matrix, regardless of order, results in the same matrix.<\/p>\n To round off this how to do matrix multiplication page we’ll display a short list of properties of matrix multiplication as a summary.<\/p>\n
\nBut it’s usually quite straightforward when we just want to multiply a matrix by a single number.<\/p>\n
\n2<\/font> × \\begin{bmatrix} 4 & 1 & 2 \\\\ 3 & 5 & 1 \\end{bmatrix}<\/span> = \\begin{bmatrix} 2\\times4 & 2\\times1 & 2\\times2 \\\\ 2\\times3 & 2\\times5 & 2\\times1 \\end{bmatrix}<\/span> = \\begin{bmatrix} 8 & 2 & 4 \\\\ 6 & 10 & 2 \\end{bmatrix}<\/span>\n
\n
\nThings are a little bit more complex when we want to multiply matrices together however.<\/p>\n
\n<\/a>\n\n\n\nMultiplying Matrices Examples<\/span>
Same Size Matrix<\/span><\/h2>\n\n\n\n
\n= \\begin{bmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\\\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \\end{bmatrix}<\/span> \n
\nNow to see an example using numbers, with .<\/b> representing multiplication, this looks like:\n
\n\\begin{bmatrix} 1 & 3 \\\\ {\\text{-}}4 & 1 \\end{bmatrix}<\/span> × \\begin{bmatrix} 5 & 2 \\\\ 1 & 3 \\end{bmatrix}<\/span>\n
\n= \\begin{bmatrix} 1.5+3.1 & 1.2+3.3 \\\\ {\\text{-}}4.5+1.1 & {\\text{-}}4.2+1.3 \\end{bmatrix}<\/span> = \\begin{bmatrix} 8 & 11 \\\\ {\\text{-}}19 & {\\text{-}}5 \\end{bmatrix}<\/span>\n
\n
\nBut generally the size of the new matrix resulting from the multiplication, depends on how many rows and columns are in the matrices being multiplied. \n
\nAlong with the fact that matrices can only be multiplied together if one matrix has the same number of columns as the other does rows.\n
\nAs such square matrices of the same size can always be multiplied together.<\/p>\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nMatrix Order of Multiplication<\/span><\/h2>\n\n\n\n
\nAs if you have two matrices A<\/font> and B<\/font>, generally A×B<\/font> ≠ B×A<\/font>.\n
\nWe can look at the multiplication of two different 2 x 2<\/font> matrices together.<\/p>\n\\begin{bmatrix} 1 & 2 \\\\ 0 & 1 \\end{bmatrix}<\/span> and \\begin{bmatrix} 3 & 0 \\\\ 1 & 4 \\end{bmatrix}<\/span>\n
\n=> \\begin{bmatrix} 1 & 2 \\\\ 0 & 1 \\end{bmatrix}<\/span> × \\begin{bmatrix} 3 & 0 \\\\ 1 & 4 \\end{bmatrix}<\/span> = \\begin{bmatrix} 1.3 + 2.1 & 1.0 + 2.4 \\\\ 0.3 + 1.1 & 0.0 + 1.4 \\end{bmatrix}<\/span> = \\begin{bmatrix} 5 & 8 \\\\ 1 & 4 \\end{bmatrix}<\/span>\n
\n=> \\begin{bmatrix} 3 & 0 \\\\ 1 & 4 \\end{bmatrix}<\/span> × \\begin{bmatrix} 1 & 2 \\\\ 0 & 1 \\end{bmatrix}<\/span> = \\begin{bmatrix} 3.1 + 0.0 & 3.2 + 0.1 \\\\ 1.1 + 4.0 & 1.2 + 4.1 \\end{bmatrix}<\/span> = \\begin{bmatrix} 3 & 6 \\\\ 1 & 6 \\end{bmatrix}<\/span>\n
\n( When practicing how to do matrix multiplication it’s handy to be able to quickly check your answers.\n
\nA handy matrix multiplication calculator is available to use at the matrix reshish<\/i><\/a> website. )\n
\n<\/a>\n\n\n\n
\n\n\n\nIdentity Matrix Multiplication<\/span><\/h2>\n\n\n\n
\n
\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}<\/span> × \\begin{bmatrix} 4 & 3 \\\\ 1 & 9 \\end{bmatrix}<\/span> = \\begin{bmatrix} 4+0 & 3+0 \\\\ 0+1 & 0+9 \\end{bmatrix}<\/span>\n
\n= \\begin{bmatrix} 4 & 3 \\\\ 1 & 9 \\end{bmatrix}<\/span>\n
\n\\begin{bmatrix} 4 & 3 \\\\ 1 & 9 \\end{bmatrix}<\/span> × \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}<\/span> = \\begin{bmatrix} 4+0 & 0+3 \\\\ 1+0 & 0+9 \\end{bmatrix}<\/span>\n
\n= \\begin{bmatrix} 4 & 3 \\\\ 1 & 9 \\end{bmatrix}<\/span>\n
\nThis example was just with a simple ( 2 × 2 ) matrix, but the result will be the same with larger matrices also.\n
\n<\/a>\n\n\n\n
\n\n\n\nProperties of Matrix Multiplication<\/span><\/h2>\n\n\n\n
\n
\n1)<\/font><\/b> A<\/font> × ( BC<\/font> ) = AB<\/font> × C<\/font> ( associative property )\n
\n2)<\/font><\/b> cA<\/font> × B<\/font> = A<\/font> × cB<\/font> ( where c<\/font> is a constant )\n
\n3)<\/font><\/b> AI<\/font><\/font> = A<\/font> , I<\/font>A<\/font> = A<\/font> ( where I<\/font><\/font> is the identity matrix )\n
\n\n\n