# How to Find the Determinant of the $3\times 3$ Matrix

## Problem 138

Find the determinant of the matix
$A=\begin{bmatrix} 100 & 101 & 102 \\ 101 &102 &103 \\ 102 & 103 & 104 \end{bmatrix}.$

## Solution.

Note that the determinant does not change if the $i$-th row is added by a scalar multiple of the $j$-th row if $i \neq j$.
We use this fact about the determinant and compute $\det(A)$ as follows.
\begin{align*}
\det(A)&=\begin{vmatrix}
100 & 101 & 102 \\
101 &102 &103 \\
102 & 103 & 104
\end{vmatrix}\5 pt] &=\begin{vmatrix} 100 & 101 & 102 \\ 101 &102 &103 \\ 1 & 1 & 1 \end{vmatrix} \quad (\text{by } R_3-R_2)\\[5 pt] &=\begin{vmatrix} 100 & 101 & 102 \\ 1 &1 &1 \\ 1 & 1 & 1 \end{vmatrix} \quad (\text{by } R_2-R_1)\\[5 pt] &=\begin{vmatrix} 100 & 101 & 102 \\ 1 &1 &1 \\ 0 & 0 & 0 \end{vmatrix} \quad (\text{by } R_3-R_1)\\[5 pt] &=0 \quad (\text{by the third row cofactor expansion}.) \end{align*} Therefore the determinant \det(A) is zero. Sponsored Links ### More from my site • Compute Determinant of a Matrix Using Linearly Independent Vectors Let A be a 3 \times 3 matrix. Let \mathbf{x}, \mathbf{y}, \mathbf{z} are linearly independent 3-dimensional vectors. Suppose that we have \[A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […] • How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following n \times n matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […] • For Which Choices of x is the Given Matrix Invertible? Determine the values of x so that the matrix \[A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix} is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.   Solution. We use the fact that a matrix is invertible […]
• Rotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by $A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.$ (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
• Find All Values of $x$ so that a Matrix is Singular Let $A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.   Hint. Use the fact that a matrix is singular if and only […]
• Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam) Let $\begin{bmatrix} 0 & 0 & 1 \\ 1 &0 &0 \\ 0 & 1 & 0 \end{bmatrix}.$ (a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers? (b) Calculate $A^{2009}$. (Princeton University, […]
• Maximize the Dimension of the Null Space of $A-aI$ Let $A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.$ Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
• Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable Determine all eigenvalues and their algebraic multiplicities of the matrix $A=\begin{bmatrix} 1 & a & 1 \\ a &1 &a \\ 1 & a & 1 \end{bmatrix},$ where $a$ is a real number.   Proof. To find eigenvalues we first compute the characteristic polynomial of the […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less

Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let \[U=\{...

Close