--- title: "How to Find the Determinant and Inverse Matrix (2×2 and 3×3) with Examples" url: https://modsengineeringmath.com/en/blog/determinant-and-inverse-matrix/ site: "Mod's Engineering Mathematics (อาจารย์มด)" language: en updated: 2026-10-16 --- > Learn to find 2×2 and 3×3 determinants by cofactors and the inverse matrix with the adjugate formula, with numeric examples checked by multiplying back, and the condition for a singular matrix. Markdown copy of https://modsengineeringmath.com/en/blog/determinant-and-inverse-matrix/ for AI assistants and search tools. Math is LaTeX ($…$ and $$…$$); interactive labs and figures are only on the web page. # How to Find the Determinant and Inverse Matrix (2×2 and 3×3) with Examples THภาษาไทยENEnglish // Article · Linear algebra The determinant tells you whether a matrix is invertible, and the inverse is used to solve systems of equations such as the currents in a multi-loop circuit. This article walks through the hand calculations with examples checked by multiplying back. 30-second summary - 2×2: $\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$ - 3×3: expand along the first row with cofactors (alternating signs $+\,-\,+$) - Inverse: $A^{-1}=\dfrac{1}{\det A}\operatorname{adj}(A)$, which exists only when $\det A\ne0$ ## The 2×2 determinant $$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$$ Example: $\det\begin{pmatrix}4&7\\2&6\end{pmatrix}=4\cdot6-7\cdot2=24-14=10$ ## The 3×3 determinant by cofactors Expand along the first row: multiply each entry by the 2×2 determinant that remains when its row and column are removed, with alternating signs $+,-,+$. $$\det A=a_{11}M_{11}-a_{12}M_{12}+a_{13}M_{13}$$ Example ### $A=\begin{pmatrix}2&1&3\\0&-1&4\\1&2&5\end{pmatrix}$ - $$\det A=2\begin{vmatrix}-1&4\\2&5\end{vmatrix}-1\begin{vmatrix}0&4\\1&5\end{vmatrix}+3\begin{vmatrix}0&-1\\1&2\end{vmatrix}$$ - $$=2(-5-8)-1(0-4)+3(0+1)=-26+4+3$$ $\det A=-19$ (not zero, so the inverse exists) For 3×3 you can also use the Sarrus rule: copy the first two columns to the right, add the products along the three downward diagonals and subtract the products along the three upward diagonals. The result is the same. ## The 2×2 inverse (shortcut formula) $$\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$ Swap $a$ and $d$, change the signs of $b$ and $c$, then divide by the determinant. Example ### $N=\begin{pmatrix}4&7\\2&6\end{pmatrix}$ - $$N^{-1}=\frac{1}{10}\begin{pmatrix}6&-7\\-2&4\end{pmatrix}=\begin{pmatrix}0.6&-0.7\\-0.2&0.4\end{pmatrix}$$ - Check: the first row of $N$ times the first column of $N^{-1}$ $$4(0.6)+7(-0.2)=2.4-1.4=1$$ And $4(-0.7)+7(0.4)=0$, $2(0.6)+6(-0.2)=0$, $2(-0.7)+6(0.4)=1$, so $NN^{-1}=I$ ✓ ## The 3×3 inverse with the adjugate Steps: (1) find the cofactor matrix $C$; (2) swap rows and columns (transpose) to get $\operatorname{adj}(A)=C^{T}$; (3) divide by $\det A$. Continuing the example $A$ above ### Find $A^{-1}$ - Cofactors $C_{ij}=(-1)^{i+j}M_{ij}$ $$C=\begin{pmatrix}-13&4&1\\1&7&-3\\7&-8&-2\end{pmatrix}$$ - Transpose $$\operatorname{adj}(A)=C^{T}=\begin{pmatrix}-13&1&7\\4&7&-8\\1&-3&-2\end{pmatrix}$$ - Divide by $\det A=-19$ $$A^{-1}=\frac{1}{19}\begin{pmatrix}13&-1&-7\\-4&-7&8\\-1&3&2\end{pmatrix}$$ Check: the first row of $A$ times the first column of $A^{-1}$: $\dfrac{1}{19}\big(2(13)+1(-4)+3(-1)\big)=\dfrac{19}{19}=1$ ✓ and the first row times the second column: $\dfrac{1}{19}\big(2(-1)+1(-7)+3(3)\big)=0$ ✓ ## What it is used for - Solving systems of equations: $A\mathbf{x}=\mathbf{b}\Rightarrow\mathbf{x}=A^{-1}\mathbf{b}$, for example the loop currents in a multi-loop electric circuit. - Checking for a unique solution: if $\det A\ne0$ the system has exactly one solution. - Eigenvalues: found from $\det(A-\lambda I)=0$, used in the stability analysis of state-space systems. ## Keep learning in the library The full lessons are in Thai, with English technical terms and the same formulas. - Determinants: properties, cofactors and row operations (in Thai) - The inverse matrix, by both the adjugate and Gauss–Jordan (in Thai) - Matrix lab: a calculator that shows the working step by step (in Thai) - Applications in electric circuits and power systems (in Thai) ## Frequently asked questions What does a zero determinant mean? The matrix is singular and has no inverse, and the system $A\mathbf{x}=\mathbf{b}$ does not have a unique solution: it has either no solution or infinitely many. The rows or columns of the matrix are linearly dependent. Can the Sarrus rule be used for a 4×4 matrix? No. The Sarrus rule (drawing diagonals) works only for 3×3. Larger matrices use cofactor expansion or row operations down to a triangular matrix. Why do we divide by the determinant when finding the inverse? Because $A\,\operatorname{adj}(A)=\det(A)\,I$, so we must divide by $\det(A)$ to get the identity matrix. That is impossible when $\det(A)=0$. Which method is preferred for finding an inverse in practice? By hand, use the shortcut formula for 2×2, and the adjugate or row operations (Gauss–Jordan) for 3×3. In computer programs one usually solves $A\mathbf{x}=\mathbf{b}$ by factorisation without forming the inverse explicitly, for better accuracy. ## More articles worth reading Linear algebra Cramer's Rule: How to Use It, Why It Works, and When Not to Use It Cramer's rule explained: replace a column by the constants and divide determinants, 2×2 and 3×3 examples, one unknown, a two-line proof, the Δ = 0… 25 Oct 2026Read the article → Linear algebra 12 Determinant Properties That Appear on Every Exam: det(kA), det(A⁻¹), det(adj A) and the Traps The 12 determinant properties used in entrance exams and engineering mathematics: row operations, det(kA) = kⁿ det A, products, inverses and the… 25 Oct 2026Read the article → Linear algebra Learning Matrices and Determinants: A 7-Step Roadmap from High School and Entrance Exams to Engineering Mathematics A roadmap for matrices, determinants, rank, Cramer's rule and numerical methods: what to learn first, how far entrance exams go, what university… 25 Oct 2026Read the article → All 7 articles in Vectors & Linear Algebra → · All articles ## Frequently asked questions **What does a zero determinant mean?** The matrix is singular and has no inverse, and the system $A\mathbf{x}=\mathbf{b}$ does not have a unique solution: it has either no solution or infinitely many. The rows or columns of the matrix are linearly dependent. **Can the Sarrus rule be used for a 4×4 matrix?** No. The Sarrus rule (drawing diagonals) works only for 3×3. Larger matrices use cofactor expansion or row operations down to a triangular matrix. **Why do we divide by the determinant when finding the inverse?** Because $A\,\operatorname{adj}(A)=\det(A)\,I$, so we must divide by $\det(A)$ to get the identity matrix. That is impossible when $\det(A)=0$. **Which method is preferred for finding an inverse in practice?** By hand, use the shortcut formula for 2×2, and the adjugate or row operations (Gauss–Jordan) for 3×3. In computer programs one usually solves $A\mathbf{x}=\mathbf{b}$ by factorisation without forming the inverse explicitly, for better accuracy.