Linear Algebra Examples

Solve Using an Inverse Matrix
,
Find the from the system of equations.
Find the inverse of the coefficient matrix of .
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The inverse of a matrix can be found using the formula where is the determinant of .
If then
The determinant of is .
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These are both valid notations for the determinant of a matrix.
The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by to get .
Multiply by to get .
Add and to get .
Substitute the known values into the formula for the inverse of a matrix.
Simplify each element of the matrix .
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Simplify element by multiplying to get .
Simplify element by multiplying to get .
Multiply by each element of the matrix.
Simplify each element of the matrix .
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Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Left multiply both sides of the matrix equation by the inverse matrix .
Any matrix multiplied by its inverse is equal to all the time. . .
Simplify the right side of the equation.
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Multiply each row in the first matrix by each column in the second matrix .
Simplify each element of the matrix by multiplying out all the expressions.
The equation after simplifying the right and left side of the equation is .
Find the solution.
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