{"id":2293,"date":"2016-11-03T20:04:12","date_gmt":"2016-11-03T20:04:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&p=2293"},"modified":"2017-04-04T19:55:19","modified_gmt":"2017-04-04T19:55:19","slug":"summary-solve-systems-with-inverses","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/summary-solve-systems-with-inverses\/","title":{"raw":"Summary: Solve Systems With Inverses","rendered":"Summary: Solve Systems With Inverses"},"content":{"raw":"
Key Equations<\/h2>\r\n
<\/colgroup>\r\n
\r\n
\r\n
Identity matrix for a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\r\n
Multiplicative inverse of a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\r\n
[latex]{A}^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{cc}d& -b\\\\ -c& a\\end{array}\\right],\\text{ where }ad-bc\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Key Concepts<\/h2>\r\n
\r\n \t
An identity matrix has the property [latex]AI=IA=A[\/latex].<\/li>\r\n \t
An invertible matrix has the property [latex]A{A}^{-1}={A}^{-1}A=I[\/latex].<\/li>\r\n \t
Use matrix multiplication and the identity to find the inverse of a [latex]2\\times 2[\/latex] matrix.<\/li>\r\n \t
The multiplicative inverse can be found using a formula.<\/li>\r\n \t
Another method of finding the inverse is by augmenting with the identity.<\/li>\r\n \t
We can augment a [latex]3\\times 3[\/latex] matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse.<\/li>\r\n \t
Write the system of equations as [latex]AX=B[\/latex], and multiply both sides by the inverse of [latex]A:{A}^{-1}AX={A}^{-1}B[\/latex].<\/li>\r\n \t
We can also use a calculator to solve a system of equations with matrix inverses.<\/li>\r\n<\/ul>\r\n
Glossary<\/h2>\r\nidentity matrix<\/strong> a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra\r\n\r\nmultiplicative inverse of a matrix<\/strong> a matrix that, when multiplied by the original, equals the identity matrix","rendered":"
Key Equations<\/h2>\n
\n
\n
\n
<\/colgroup>\n
\n
\n
Identity matrix for a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\n
Multiplicative inverse of a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\n
[latex]{A}^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{cc}d& -b\\\\ -c& a\\end{array}\\right],\\text{ where }ad-bc\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Key Concepts<\/h2>\n
\n
An identity matrix has the property [latex]AI=IA=A[\/latex].<\/li>\n
An invertible matrix has the property [latex]A{A}^{-1}={A}^{-1}A=I[\/latex].<\/li>\n
Use matrix multiplication and the identity to find the inverse of a [latex]2\\times 2[\/latex] matrix.<\/li>\n
The multiplicative inverse can be found using a formula.<\/li>\n
Another method of finding the inverse is by augmenting with the identity.<\/li>\n
We can augment a [latex]3\\times 3[\/latex] matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse.<\/li>\n
Write the system of equations as [latex]AX=B[\/latex], and multiply both sides by the inverse of [latex]A:{A}^{-1}AX={A}^{-1}B[\/latex].<\/li>\n
We can also use a calculator to solve a system of equations with matrix inverses.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
identity matrix<\/strong> a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra<\/p>\n
multiplicative inverse of a matrix<\/strong> a matrix that, when multiplied by the original, equals the identity matrix<\/p>\n\n\t\t\t \n\t\t\t