Content

• Steps
• Advice
• Warning

Inversion is often used in calculus to simplify problematic problems in other ways. For example, it is easier to multiply with the inverse of a fraction than it is to directly divide it by that number. This is the inverse. Likewise, since there are no fraction signs for the matrix, you will have to multiply its inverse matrix. Calculating the inverse matrix of a 3x3 matrix can be very tedious, but it is a problem worth considering. You can also use an advanced graphing calculator to do this.

Steps

Method 1 of 3: Create an additional matrix to find the inverse matrix

1. 

   Check the determinant of the matrix. The first step: find the determinant of the matrix. If the determinant is 0, that's done: this matrix is ​​not reversible. The determinant of a matrix M can be denoted det (M).
   • To find the inverse of a 3x3 matrix, you must first calculate its determinant.
   • To review how to find the determinant of matrices, refer to the article Find the 3x3 matrix determinant.

2. 

   
   
   Original matrix transposition. Transposition means reflecting the matrix across the main diagonal, or in other words, swapping th element (i, j) and element (j, i). When transpose elements of a matrix, the main diagonal (running from the upper left corner to the lower right corner) remains constant.
   • Another way to understand transposition is that you will rewrite the matrix so that the first row becomes the first column, the middle row becomes the middle column, and the third row becomes the third column. Notice the color elements in the illustration above and notice the new position of the numbers.

3. 

   
   
   Find the determinant of each 2x2 sub-matrix. All elements of the new 3x3 displacement matrix are linked to a corresponding 2x2 'sub' matrix. To find the sub-matrix of each element, first highlight the row and column of the first element. All 5 elements will be highlighted. The remaining four elements form the sub-matrix.
   • In the above example, if you want to find the sub-matrix of the element in row two, column one, you highlight five word sections in the second row and first column. The remaining four elements are the corresponding sub-matrix.
   • Find the determinant of each sub-matrix by multiplying diagonally and subtracting two products from each other, as shown in the figure above.
   • Read more to learn more about sub-matrices and their uses.

4. 

   
   
   Make a matrix of algebraic subsections. Place the result obtained from the previous step into a new matrix composed of algebraic subsections by placing each sub-matrix identifier in the corresponding position in the original matrix. Thus, the determinant calculated from the element (1,1) of the original matrix will be placed at position (1,1). Next, you will have to change the replacement sign of this new matrix according to the reference table shown in the illustration above.
   • When determining the sign, the mark of the first molecule of the leading is kept. The second element's sign is reversed. The sign of the third element is preserved. Continue like that for the rest of the matrix. Note that the sign (+) or (-) in the reference chart does not indicate that until the end, the element will carry positive or negative signs. They only show that the elements will be kept intact (+) or changed with (-).
   • Refer to matrix basics for more on algebraic appendages.
   • The final result that we get in this step is the complementary matrix of the original matrix. It is sometimes also called a conjugate matrix and is denoted Adj (M).

5. 

   Divide all elements of the complement matrix by the determinant. Use the determinant of the matrix M you calculated in the first step (to check if the matrix is ​​reversible). Now divide every element of the matrix by this value. Put the quotient of each division into the position of the original element, and we get the inverse matrix of the original matrix.
   • The sample matrix presented in the illustration has determinant of 1. Therefore, when we divide every element of the complementary matrix by the determinant, we obtain itself (you won't always be so lucky). .
   • Instead of dividing, some documentation demonstrates this step as multiplying every element of M by 1 / det (M). Mathematically, they are equivalent.
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Method 2 of 3: Decrease the linear row to find the inverse matrix

1. 

   Add the unit matrix to the original matrix. Write the base matrix M, draw a vertical line to the right of that matrix, and then write the unit matrix to the right of this line. At this point, we have a matrix with three rows and six columns.
   • Remember that the identity matrix is ​​a special matrix with all elements on the main diagonal, running from the top left corner to the lower right corner, equal to 1 and all elements in the remaining positions equal to zero.

2. 

   Perform a linear row reduction. The goal here is to create the unit matrix in the left part of the newly expanded matrix. When performing the row reduction steps on the left, you must do the corresponding part on the right - the part that is your unit matrix.
   • Remember that row reduction is performed as a combination of scalar multiplication and row addition or subtraction, in order to isolate individual elements of the matrix.

3. 

   Continue until the unit matrix is ​​formed. Continue the linear reduction until the identity matrix appears (elements on the diagonal are equal to 1, other elements are equal to 0) in the left part of the expanded matrix. Once this step is reached, the right part of the vertical divider is the inverse matrix of the original matrix.

4. 

   Rewrite the inverse matrix. Duplicate the elements that are currently on the right part of the vertical divider and that is your inverse matrix. advertisement

Method 3 of 3: Find the inverse matrix with pocket calculator

1. 

   Choose a calculator that can solve matrices. A simple four-function calculator won't be able to find the inverse matrix directly for you. However, because of mathematical repetition, an advanced graphing calculator, such as the Texas Instruments TI-83 or TI-86, can greatly reduce your work to do.

2. 

   Enter the matrix into the calculator. First, enter the Matrix function of your calculator by pressing the Matrix key, if it is available on your device. With the Texas Instruments machine, you will have to press 2 Matrix.

3. 

   Select the Edit submenu. To access this submenu, you may need to use the arrow buttons or select the appropriate function keys located in the top row of the computer keyboard, depending on its design.

4. 

   Choose a name for your matrix. Most calculators are equipped to work with 3 to 10 matrices, lettered names, A through J. Normally, let's start with. Press the Enter key to confirm the name selection.

5. 

   Enter the matrix size. This article focuses on 3x3 matrices. However, pocket calculators can handle larger matrices. Enter the number of rows, press Enter, then type the column number and press Enter.

6. 

   Enter each element of the matrix. A matrix will be displayed on the computer screen. If you have worked with the matrix function before, the matrix that you worked with before will appear on the screen. The cursor will mark the first element of the matrix. Enter the matrix value you want to solve and press Enter. The cursor will automatically move to the next element, overwriting any previous values.
   • If you want to enter negative numbers, use your calculator's negative (-) button, not the minus key. The matrix function will not read correctly.
   • If needed, you can use the arrow keys on your calculator to move through the matrix.

7. 

   Exits the matrix function. After you have entered the entire matrix value, press the Quit - Exit key (or 2 Quit, if necessary). Thanks to that, you exit the Matrix function and return to the main display screen of the calculator.

8. 

   Use the inverse key to find the inverse matrix. First, re-open the Matrix function and use the Names button to select the matrix name you used to give your matrix (maybe it is). Next, press the calculator's inverse key,. Depending on your device, you may need to use button 2. The display screen appears. Press Enter, and the inverse matrix will appear on your screen.
   • Do not use the ^ button on your computer when trying to enter A ^ -1 with individual clicks. Computers won't understand this math.
   • If you get an error message when you press the inverse key, it's more likely that your parent matrix is ​​not reversible. Maybe you should go back and be qualitative to determine if that's the cause of the error.

9. 

   Convert the inverse matrix into the correct answer. The first result returned by the computer is displayed in decimal. That is not necessarily the "correct" answer for most purposes. You should convert this decimal answer to a fraction if needed (if lucky enough, all your results are integers. However, it is very rare).
   • Maybe your calculator has a function that automatically converts decimals to fractions. For example, when using TI-86, you can go to Math function, select Misc then Frac and press Enter. Decimals will be automatically represented as fractions.
10. Most graphing calculators have square brackets (for TI-84, that is 2nd + x and 2nd + -) that allow you to enter a matrix without using a matrix function. Note: A calculator might not format a matrix until the enter / equal key is used (meaning everything will be on the same row and not very nice). advertisement

Advice

• You can follow these steps to find the inverse of a matrix that not only contains numbers, but also variables, unknowns, or even algebraic expressions.
• Write down all the steps because finding the inverse of a 3x3 matrix just by doing math is extremely difficult.
• There are calculator programs that help you to find inverse matrices, up to and including 30x30 matrices.
• Regardless of the method used, check the accuracy of the result by multiplying M by M. You will confirm that M * M = M * M = I. Where, I is the unit matrix , is composed of elements 1 located along the main diagonal and zeros elsewhere. If you do not get such results, you must have gone wrong somewhere.

Warning

• Not all 3x3 matrices have inverse matrices. If the determinant is 0, that matrix is ​​not reversible (Note that in the formula, we divide by det (M). Dividing by zero is an undefined operation).