Period+6+-+Inverse+2x2

=__**Applying Matrices to Linear System:** Inverse of a 2x2 matrix __=

//Sarah Choi, Kim Wang//

**Background Information **
It is possible to add, subtract, and multiply matrices; however, it is not possible to divide them. Matrices instead have inverses. Befor we even begin to discuss matrices, let us first familiarize ourselves with the concept of the inverse by looking at a basic algebraic equation. When working with basic algebraic equations, you can simply multiply a number by its (the reciprocal) to get 1; this result is called a multiplicative identity. Now consider the algebraic expression: By subtracting 3 from both sides, the equation can be rewritten as:



You can then multiply by sides by the reciprocal fraction 1/5 to find the value of x (in this case, x = 3).

However, in the case of matrices, you cannot divide.

Now, given the following matrix equation:



you must use the inverse of matrix A (denoted A -1 ) in order to find the value of X. Since, by definition, the product of the inverse of a number and the number is 1,



the product of the inverse of A, A -1, and X is 1X, so,



The above statement is true because when any matrix is multiplied by its inverse, it becomes a identity matrix. By definition, any matrix X multiplied by an identity matrix is equal to X.

Identity matrix I pd. 8 or Identity matrix pd. 6 Here are some examples: or:

"So... how do we do it?"- How to Find the Inverse of a 2x2 Matrix
Consider the standard matrix A:



If we insert 1, 2, 3, 4 as example numbers into the matrix, Matrix A becomes:



Steps:
__1. Find Determinant.__

__2. Change the positions of a and d__.



Plugging in the numbers from the above example, we get the following:



__3. Switch the signs of b and c.__



__4. Multiply with the reciprocal of the determinant__





5. Reduce the fractions to get your final answer:

In order to verify your answer, you can multiply the two matrices to see the final product. The final product should be an identity matrix.

To see how to multiply matrices, click here.

//* Important things to note:// 1. Non-square matrices do not have inverses. 2. Not all square matrices have inverses! Square matrices that can be inverted are known as invertible or nonsingular. On the other hand, those that cannot be inverted are known as noninvertible or singular. 3. Be careful of the order of the matrices! AB **does not** equal BA! With this in mind, be aware of the specific order of the matrices when multiplying.
 * Verifying**

To see more information on solving a system of equations with an inverse matrix, click here.

**Now you try it! ** Try these following examples. Check your answers on the bottom of the page.

//Example #1://

//Example #2://

//Example #3://

ANSWERS (Drag your mouse across the black rectangles to check your answers). Example #1:

(1/28 3/28) (-2/7 1/7)

Example #2: (-5 -3) (-3 -2)

Example #3: (-9/44 -1/88) (1/22 5/44)

How to do the inverse of 2x2 matrixes: media type="custom" key="13954882" align="center"
 * Still confused? Here are some helpful video tutorials: **

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How to find the determinant of a 2x2 matrix with cross multiplication: media type="custom" key="13955008" align="center"

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Challenge - Find the Inverse of a 3x3 Matrix
Click [|here]to find the inverse of this matrix using the identity matrix.

References: []