Solve ([5,3,7],[1,-3,0],[5,3,8]) | 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

$$\left[\begin{matrix} 5 & 3 & 7 \\1 & -3 & 0 \\5 & 3 & 8 \end{matrix}\right]$$

Calculate Determinant

$-18$

Steps Using Sarrus' Rule

Steps Using Expansion by Minors

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}5&3&7\\1&-3&0\\5&3&8\end{matrix}\right))$$

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

$$\left(\begin{matrix}5&3&7&5&3\\1&-3&0&1&-3\\5&3&8&5&3\end{matrix}\right)$$

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

$$5\left(-3\right)\times 8+7\times 3=-99$$

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

$$5\left(-3\right)\times 7+8\times 3=-81$$

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

$$-99-\left(-81\right)$$

Subtract $-81$ from $-99$.

$$-18$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}5&3&7\\1&-3&0\\5&3&8\end{matrix}\right))$$

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.

$$5det(\left(\begin{matrix}-3&0\\3&8\end{matrix}\right))-3det(\left(\begin{matrix}1&0\\5&8\end{matrix}\right))+7det(\left(\begin{matrix}1&-3\\5&3\end{matrix}\right))$$

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

$$5\left(-3\right)\times 8-3\times 8+7\left(3-5\left(-3\right)\right)$$

Simplify.

$$5\left(-24\right)-3\times 8+7\times 18$$

Add the terms to obtain the final result.

$$-18$$

Evaluate

$\left(\begin{matrix}5&3&7\\1&-3&0\\5&3&8\end{matrix}\right)$

Invert Matrix

$\left(\begin{matrix}\frac{4}{3}&\frac{1}{6}&-\frac{7}{6}\\\frac{4}{9}&-\frac{5}{18}&-\frac{7}{18}\\-1&0&1\end{matrix}\right)$

Calculate Trace

$10$

Transpose Matrix

$\left(\begin{matrix}5&1&5\\3&-3&3\\7&0&8\end{matrix}\right)$

Matrix Size

$3,3$

Reduce Matrix

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

Adjoint Matrix

$\left[\begin{array}{rrr}-24 & -3 & 21 \\ -8 & 5 & 7 \\ 18 & 0 & -18\end{array}\right]$

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 5

$A_{11}=\left|\begin{array}{rr}-3 & 0 \\ 3 & 8\end{array}\right|=-24$

Cofactor of 3

$A_{12}=-\left|\begin{array}{rr}1 & 0 \\ 5 & 8\end{array}\right|=-8$

Cofactor of 7

$A_{13}=\left|\begin{array}{rr}1 & -3 \\ 5 & 3\end{array}\right|=18$

Cofactor of 1

$A_{21}=-\left|\begin{array}{rr}3 & 7 \\ 3 & 8\end{array}\right|=-3$

Cofactor of -3

$A_{22}=\left|\begin{array}{rr}5 & 7 \\ 5 & 8\end{array}\right|=5$

Cofactor of 0

$A_{23}=-\left|\begin{array}{rr}5 & 3 \\ 5 & 3\end{array}\right|=0$

Cofactor of 5

$A_{31}=\left|\begin{array}{rr}3 & 7 \\ -3 & 0\end{array}\right|=21$

Cofactor of 3

$A_{32}=-\left|\begin{array}{rr}5 & 7 \\ 1 & 0\end{array}\right|=7$

Cofactor of 8

$A_{33}=\left|\begin{array}{rr}5 & 3 \\ 1 & -3\end{array}\right|=-18$

The cofactor matrix of A is [Aij]

$\left[\begin{array}{rrr}-24 & -8 & 18 \\ -3 & 5 & 0 \\ 21 & 7 & -18\end{array}\right]$

The transpose of Aij

$\left[\begin{array}{rrr}-24 & -3 & 21 \\ -8 & 5 & 7 \\ 18 & 0 & -18\end{array}\right]$