How to Find the Inverse of a 3×3 . Matrix

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The inverse is often used in calculus to simplify difficult problems in other ways. For example, it is easier to multiply by the reciprocal of a fraction than to divide by the number directly. This is the inverse. Similarly, since there is no fractional sign for the matrix, you will have to multiply by its inverse. Manually calculating the inverse of a 3×3 matrix can be tedious, but it’s a problem worth considering. You can also use an advanced graphing calculator to do this.

Table of Contents

Create an additional matrix to find the inverse

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Check the determinant of the matrix. First step: find the determinant of the matrix. If the determinant is 0, the job is done: the matrix is not invertible. The determinant of the matrix M can be denoted det(M). ^{[1] X Research Source}

To find the inverse of a 3×3 matrix, you must first calculate its determinant.
To review how to find the determinant of a matrix, refer to the article Finding the determinant of a 3×3 matrix.

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Transpose the original matrix. Transpose means mirror the matrix through the main diagonal, or in other words, swap the (i,j) and (j,i)th element. When transposing the elements of a matrix, the principal diagonal (running from the upper left corner to the lower right corner) remains constant. ^{[2] X Research Source}

Another way of understanding transposition is that you would 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 factor in the illustration above and notice the new placement of the numbers.

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Find the determinant of each 2×2 submatrix. Every element of the new 3×3 transpose matrix is associated with a corresponding 2×2 “sub” matrix. To find the submatrix 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 submatrix. ^{[3] X Research Sources}

In the above example, if you want to find the submatrix of the element located in row two, column one, you highlight five parts from the second row and first column. The remaining four elements are the corresponding submatrix.
Find the determinant of each submatrix by multiplying diagonally and subtracting the two products, as shown in the figure above.
Read on to learn more about submatrices and their uses.

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Create a matrix of algebraic subsections. Place the result obtained from the previous step into a new matrix made up of algebraic subdivisions by placing each submatrix determinant in its 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 sign of the replacement element of this new matrix according to the reference table shown in the illustration above. ^{[4] X Research Sources}

When specifying the sign, the sign of the first element of the leading row will be preserved. The sign of the second element is inverted. The sign of the third element is preserved. Continue in the same way for the rest of the matrix. Note that the (+) or (-) signs in the reference chart do not indicate whether the element will have a positive or negative sign at the end. They just indicate that the elements will stay the same (+) or change the sign (-).
Refer to matrix basics for more on algebraic subsection.
The final result that we get in this step is the addition matrix of the original matrix. It is sometimes called the conjugate matrix and is denoted by Adj(M).

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Divide every element of the matrix by the determinant. Use the determinant of the matrix M that you calculated in the first step (to check if the matrix is invertible). Now, divide every element of the matrix by this value. Putting the quotient of each division in place of the original element, we get the inverse matrix of the original matrix. ^{[5] X Research Sources}

The sample matrix presented in the illustration has a determinant of 1. Thus, dividing every element of the matrix by the determinant, we get itself (you won’t always be so lucky) .
Instead of division, some literature presents this step as multiplying every element of M by 1/det(M). Mathematically, they are equivalent.

Linear row reduction to find the inverse matrix

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Add the unit matrix to the original matrix. Write the original matrix M, draw a vertical line to the right of that matrix and then write the unit matrix to the right of this line. Now, we have a matrix with three rows and six columns. ^{[6] X Research Sources}

Remember that the unit matrix is a special matrix where every element on the main diagonal, running from the upper left corner to the lower right corner, is equal to 1 and all elements in the remaining positions are equal to 0.

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Perform linear row reduction. The goal here is to create a unit matrix in the left part of the newly expanded matrix. When you do the row reduction steps on the left, you must do the same on the right side, which is your unit matrix. ^{[7] X Research Sources}

Remember that row reduction is performed as a combination of scalar multiplication and row addition or subtraction, to isolate the individual elements of the matrix.

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Continue until the unit matrix is formed. Continue the linear row reduction until a unit matrix appears (elements on the diagonal equal 1, all other elements equal 0) in the left part of the expanded matrix. Once this step is reached, the right part of the vertical division is the inverse of the original matrix. ^{[8] X Research Sources}

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Rewrite the inverse matrix. Duplicate the elements that are currently on the right part of the vertical division and that is your inverse matrix. ^{[9] X Research Source}

Find the inverse matrix using a pocket calculator

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Choose a calculator capable of solving matrices. The simple four-function calculator won’t be able to directly find the inverse matrix for you. However, because of the repetitiveness of the math operations, an advanced graphing calculator, such as the Texas Instruments TI-83 or TI-86, can greatly reduce your workload. ^{[10] X Research Source}

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Enter the matrix into the calculator. First, enter your calculator’s Matrix function by pressing the Matrix key, if this key is available on your machine. With the Texas Instruments machine, you will have to hit 2 ^nd Matrix.

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Select the Edit submenu. To access this submenu, you may have to use the arrow buttons or select the appropriate function keys located in the upper row of the computer keyboard, depending on its design. ^{[11] X Research Source}

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Choose a name for your matrix. Most calculators are equipped to work with 3 to 10 matrices, named with letters, A through J. Usually, just start with [A]. Press the Enter key to confirm the name selection. ^{[12] X Research Source}

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Enter the matrix size. This article focuses on 3×3 matrices. However, pocket computers can handle larger matrices. Type the row number, press Enter, then enter the column number, and press Enter. ^{[13] X Research Source}

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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 you worked with will appear on the screen. The pointer will mark the first element of the matrix. Enter the matrix value that you want to solve and press Enter. The pointer will automatically move to the next element, overwriting any previous values. ^{[14] X Research Source}

If you want to enter a negative number, use the negative (-) button of the calculator, not the minus key. The matrix function will not read properly.
If needed, you can use the arrow keys on your calculator to move through the matrix.

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Exit the matrix function. After you have entered all the values of the matrix, press the Quit key – Exit (or 2 ^nd Quit, if necessary). With this, you exit the Matrix function and return to the main display of the calculator. ^{[15] X Research Source}

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Use the inverse key to find the inverse matrix. First, reopen the Matrix function and use the Names button to select the matrix name that you used to give your matrix (probably [A]). Next, press the calculator’s inverse key,

x

-

first

{displaystyle x^{-1}}

x^{{-1}}

. Depending on the device, you may have to use the ^2nd button. The display appears

A

-

first

{displaystyle A^{-1}}

A^{{-1}}

. Press Enter, and the inverse matrix will appear on your screen. ^{[16] X Research Source}

Don’t use the ^ button on a calculator when trying to enter A^-1 with individual presses. The computer will not understand this math.
If you get an error message when you press the inverse key, chances are your original matrix is not invertible. Maybe you should go back and compute the determinant to determine if that’s the reason for the error.

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Convert the inverse matrix to the correct answer. The first result returned by the calculator is represented as a decimal. That’s not necessarily the “correct” answer for most uses. You should convert this decimal answer to a fraction if necessary (if you’re lucky enough, all your results will be integers. However, that’s very rare). ^{[17] X Research Source}

Maybe your calculator has a function to automatically convert decimals to fractions. For example, when using the TI-86, you can go into the Math function, select Misc then Frac and press Enter. Decimals are automatically represented as fractions.

Advice

You can follow these steps to find the inverse of a matrix that contains not only numbers but also variables, unknowns, or even algebraic expressions.
Write down all the steps because finding the inverse of a 3×3 matrix just by doing math is extremely difficult.
There are computer programs that help you find the inverse matrix ^{[18] X Research Source} , up to and including 30×30 matrices.
Whatever method is used, check the accuracy of the result by multiplying M by M ^-1 . You will find that M*M ^-1 = M ^-1 *M = I. Where, I is the unit matrix, made up of 1’s along the main diagonal and 0’s on the main diagonal. other locations. If you don’t get such results, you must have made a mistake somewhere.

Warning

Not every 3×3 matrix has an inverse. If the determinant is 0, the matrix is not invertible (Note that in the formula we divide by det(M).Dividing by zero is an undefined operation).

Steps

Create an additional matrix to find the inverse

Linear row reduction to find the inverse matrix

Find the inverse matrix using a pocket calculator

Advice

Warning

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