Solve ([10,20,30],[40,50,60],[70,80,90]) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\left[\begin{matrix} 10 & 20 & 30 \\40 & 50 & 60 \\70 & 80 & 90 \end{matrix}\right]$$

Calculate Determinant

$0$

Steps Using Sarrus' Rule

Steps Using Expansion by Minors

Steps Using Sarrus' Rule

Find the determinant of the matrix using the method of diagonals.

$$det(\left(\begin{matrix}10&20&30\\40&50&60\\70&80&90\end{matrix}\right))$$

Extend the original matrix by repeating the first two columns as the fourth and fifth columns.

$$\left(\begin{matrix}10&20&30&10&20\\40&50&60&40&50\\70&80&90&70&80\end{matrix}\right)$$

Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.

$$10\times 50\times 90+20\times 60\times 70+30\times 40\times 80=225000$$

Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.

$$70\times 50\times 30+80\times 60\times 10+90\times 40\times 20=225000$$

Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.

$$225000-225000$$

Subtract $225000$ from $225000$.

$$0$$

Steps Using Expansion by Minors

Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).

$$det(\left(\begin{matrix}10&20&30\\40&50&60\\70&80&90\end{matrix}\right))$$

To expand by minors, multiply each element of the first row by its minor, which is the determinant of the $2\times 2$ matrix created by deleting the row and column containing that element, then multiply by the element's position sign.

$$10det(\left(\begin{matrix}50&60\\80&90\end{matrix}\right))-20det(\left(\begin{matrix}40&60\\70&90\end{matrix}\right))+30det(\left(\begin{matrix}40&50\\70&80\end{matrix}\right))$$

For the $2\times 2$ matrix $\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$, the determinant is $ad-bc$.

$$10\left(50\times 90-80\times 60\right)-20\left(40\times 90-70\times 60\right)+30\left(40\times 80-70\times 50\right)$$

Simplify.

$$10\left(-300\right)-20\left(-600\right)+30\left(-300\right)$$

Add the terms to obtain the final result.

$$0$$

Evaluate

$\left(\begin{matrix}10&20&30\\40&50&60\\70&80&90\end{matrix}\right)$

Calculate Trace

$150$

Transpose Matrix

$\left(\begin{matrix}10&40&70\\20&50&80\\30&60&90\end{matrix}\right)$

Matrix Size

$3,3$

Reduce Matrix

$\left(\begin{matrix}1&0&-1\\0&1&2\\0&0&0\end{matrix}\right)$

Adjoint Matrix

$\left[\begin{array}{rrr}-300 & 600 & -300 \\ 600 & -1200 & 600 \\ -300 & 600 & -300\end{array}\right]$

To find the adjoint of a matrix, first find the cofactor matrix of the given matrix. Then find the transpose of the cofactor matrix.

Cofactor of 10

$A_{11}=\left|\begin{array}{rr}50 & 60 \\ 80 & 90\end{array}\right|=-300$

Cofactor of 20

$A_{12}=-\left|\begin{array}{rr}40 & 60 \\ 70 & 90\end{array}\right|=600$

Cofactor of 30

$A_{13}=\left|\begin{array}{rr}40 & 50 \\ 70 & 80\end{array}\right|=-300$

Cofactor of 40

$A_{21}=-\left|\begin{array}{rr}20 & 30 \\ 80 & 90\end{array}\right|=600$

Cofactor of 50

$A_{22}=\left|\begin{array}{rr}10 & 30 \\ 70 & 90\end{array}\right|=-1200$

Cofactor of 60

$A_{23}=-\left|\begin{array}{rr}10 & 20 \\ 70 & 80\end{array}\right|=600$

Cofactor of 70

$A_{31}=\left|\begin{array}{rr}20 & 30 \\ 50 & 60\end{array}\right|=-300$

Cofactor of 80

$A_{32}=-\left|\begin{array}{rr}10 & 30 \\ 40 & 60\end{array}\right|=600$

Cofactor of 90

$A_{33}=\left|\begin{array}{rr}10 & 20 \\ 40 & 50\end{array}\right|=-300$

The cofactor matrix of A is [Aij]

$\left[\begin{array}{rrr}-300 & 600 & -300 \\ 600 & -1200 & 600 \\ -300 & 600 & -300\end{array}\right]$

The transpose of Aij

$\left[\begin{array}{rrr}-300 & 600 & -300 \\ 600 & -1200 & 600 \\ -300 & 600 & -300\end{array}\right]$