{"id":411,"date":"2019-07-15T22:44:50","date_gmt":"2019-07-15T22:44:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/summary-solve-systems-with-inverses\/"},"modified":"2019-07-15T22:44:50","modified_gmt":"2019-07-15T22:44:50","slug":"summary-solve-systems-with-inverses","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/summary-solve-systems-with-inverses\/","title":{"raw":"Summary: Solving Systems With Inverses","rendered":"Summary: Solving Systems With Inverses"},"content":{"raw":"\n
Key Equations<\/h2>\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 \t
An identity matrix has the property [latex]AI=IA=A[\/latex].<\/li>\n \t
An invertible matrix has the property [latex]A{A}^{-1}={A}^{-1}A=I[\/latex].<\/li>\n \t
Use matrix multiplication and the identity to find the inverse of a [latex]2\\times 2[\/latex] matrix.<\/li>\n \t
The multiplicative inverse can be found using a formula.<\/li>\n \t
Another method of finding the inverse is by augmenting with the identity.<\/li>\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>\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>\n \t
We can also use a calculator to solve a system of equations with matrix inverses.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
\n \t
\n
\n \t
identity matrix<\/strong><\/dt>\n \t
a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra<\/dd>\n<\/dl>\n
\n \t
\n
\n \t
\n
\n \t
multiplicative inverse of a matrix<\/strong><\/dt>\n \t
a matrix that, when multiplied by the original, equals the identity matrix<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n","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
\n
\n<\/dt>\n
identity matrix<\/strong><\/dt>\n
a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra<\/dd>\n<\/dl>\n
\n
\n<\/dt>\n
\n<\/dt>\n
multiplicative inverse of a matrix<\/strong><\/dt>\n
a matrix that, when multiplied by the original, equals the identity matrix<\/dd>\n<\/dl>\n\n\t\t\t \n\t\t\t