Inverse of Matrix Calculator

The calculator will find the inverse of the square matrix using the Gaussian elimination method, with steps shown.

Show Instructions
  • In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`.
  • In general, you can skip parentheses, but be very careful: e^3x is `e^3x`, and e^(3x) is `e^(3x)`.
  • Also, be careful when you write fractions: 1/x^2 ln(x) is `1/x^2 ln(x)`, and 1/(x^2 ln(x)) is `1/(x^2 ln(x))`.
  • If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
  • Sometimes I see expressions like tan^2xsec^3x: this will be parsed as `tan^(2*3)(x sec(x))`. To get `tan^2(x)sec^3(x)`, use parentheses: tan^2(x)sec^3(x).
  • Similarly, tanxsec^3x will be parsed as `tan(xsec^3(x))`. To get `tan(x)sec^3(x)`, use parentheses: tan(x)sec^3(x).
  • From the table below, you can notice that sech is not supported, but you can still enter it using the identity `sech(x)=1/cosh(x)`.
  • If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
  • All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
TypeGet
Constants
ee
pi`pi`
ii (imaginary unit)
Operations
a+ba+b
a-ba-b
a*b`a*b`
a^b, a**b`a^b`
sqrt(x), x^(1/2)`sqrt(x)`
cbrt(x), x^(1/3)`root(3)(x)`
root(x,n), x^(1/n)`root(n)(x)`
x^(a/b)`x^(a/b)`
abs(x)`|x|`
Functions
e^x`e^x`
ln(x), log(x)ln(x)
ln(x)/ln(a)`log_a(x)`
Trigonometric Functions
sin(x)sin(x)
cos(x)cos(x)
tan(x)tan(x), tg(x)
cot(x)cot(x), ctg(x)
sec(x)sec(x)
csc(x)csc(x), cosec(x)
Inverse Trigonometric Functions
asin(x), arcsin(x), sin^-1(x)asin(x)
acos(x), arccos(x), cos^-1(x)acos(x)
atan(x), arctan(x), tan^-1(x)atan(x)
acot(x), arccot(x), cot^-1(x)acot(x)
asec(x), arcsec(x), sec^-1(x)asec(x)
acsc(x), arccsc(x), csc^-1(x)acsc(x)
Hyperbolic Functions
sinh(x)sinh(x)
cosh(x)cosh(x)
tanh(x)tanh(x)
coth(x)coth(x)
1/cosh(x)sech(x)
1/sinh(x)csch(x)
Inverse Hyperbolic Functions
asinh(x), arcsinh(x), sinh^-1(x)asinh(x)
acosh(x), arccosh(x), cosh^-1(x)acosh(x)
atanh(x), arctanh(x), tanh^-1(x)atanh(x)
acoth(x), arccoth(x), cot^-1(x)acoth(x)
acosh(1/x)asech(x)
asinh(1/x)acsch(x)

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Enter the elements of the matrix or

Write all suggestions in comments below.

Solution

Your input: find the inverse of $$$A=\left[ \begin{array}{cc} 2 & 1 \\\\ 1 & 3 \end{array} \right]$$$

To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be inverse matrix.

So, augment matrix with identity:

$$$\left[ \begin{array}{cc|cc}2&1&1&0 \\\\ 1&3&0&1\end{array}\right]$$$

Make zeros in column 1 except the entry at row 1, column 1 (pivot entry).

Divide row 1 by $$$2$$$ $$$\left(R_1=\frac{R_1}{2}\right)$$$:

$$$\left[ \begin{array}{cc|cc}1&\frac{1}{2}&\frac{1}{2}&0 \\\\ 1&3&0&1\end{array}\right]$$$

Subtract row 1 from row 2 $$$\left(R_2=R_2-R_1\right)$$$:

$$$\left[ \begin{array}{cc|cc}1&\frac{1}{2}&\frac{1}{2}&0 \\\\ 0&\frac{5}{2}&- \frac{1}{2}&1\end{array}\right]$$$

Make zeros in column 2 except the entry at row 2, column 2 (pivot entry).

Multiply row 2 by $$$\frac{2}{5}$$$ $$$\left(R_2=\left(\frac{2}{5}\right)R_2\right)$$$:

$$$\left[ \begin{array}{cc|cc}1&\frac{1}{2}&\frac{1}{2}&0 \\\\ 0&1&- \frac{1}{5}&\frac{2}{5}\end{array}\right]$$$

Subtract row 2 multiplied by $$$\frac{1}{2}$$$ from row 1 $$$\left(R_1=R_1-\left(\frac{1}{2}\right)R_2\right)$$$:

$$$\left[ \begin{array}{cc|cc}1&0&\frac{3}{5}&- \frac{1}{5} \\\\ 0&1&- \frac{1}{5}&\frac{2}{5}\end{array}\right]$$$

As can be seen, we have obtained the identity matrix to the left. So, we are done.

Answer: $$$A^{-1}=\left[ \begin{array}{cc} \frac{3}{5} & - \frac{1}{5} \\\\ - \frac{1}{5} & \frac{2}{5} \end{array} \right]$$$

Decimal form: $$$A^{-1}=\left[ \begin{array}{cc} 0.6 & -0.2 \\\\ -0.2 & 0.4 \end{array} \right]$$$