9.7 Solving systems with inverses  (Page 5/8)

Key equations

Key concepts

• An identity matrix has the property $AI=IA=A.$ See [link] .
• An invertible matrix has the property $A{A}^{\mathrm{-1}}=A^{\mathrm{-1}}A=I.$ See [link] .
• Use matrix multiplication and the identity to find the inverse of a $2\times 2$ matrix. See [link] .
• The multiplicative inverse can be found using a formula. See [link] .
• Another method of finding the inverse is by augmenting with the identity. See [link] .
• We can augment a $3\times 3$ 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. See [link] .
• Write the system of equations as $AX=B,$ and multiply both sides by the inverse of $A:A^{\mathrm{-1}}AX=A^{\mathrm{-1}}B.$ See [link] and [link] .
• We can also use a calculator to solve a system of equations with matrix inverses. See [link] .

Section exercises

Verbal