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مرکز یادگیری / جبر خطی

How to find the determinant of a 2×2 or 3×3 matrix

How to find the determinant of a matrix: the ad − bc rule for 2×2, cofactor expansion for 3×3, a shortcut for triangular matrices, and what a zero determinant tells you.

2 دقیقه مطالعهبه‌روزرسانی ۱۱ آبان ۱۴۰۵نوشته Sumzo teamبازبینی: Sumzo team

The determinant is a single number worked out from a square matrix. It tells you whether the matrix has an inverse (only if the determinant isn’t 0), how a transformation scales area or volume, and whether a system of equations has exactly one solution. Here is how to find the determinant by hand for 2×2 and 3×3 matrices.

Find the determinant of a 2×2 matrix

∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

مثال حل‌شده

Easy: a 2×2 determinant

Find the determinant of [5234]\begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}.

نمایش راه‌حل
  1. Multiply down the main diagonal, then subtract the other diagonal.
    5⋅4−2⋅3=20−6=145 \cdot 4 - 2 \cdot 3 = 20 - 6 = 14

پاسخ: 1414

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How to find the determinant of a 3×3 matrix

∣abcdefghi∣=a(ei−fh)−b(di−fg)+c(dh−eg)\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)
Cofactor expansion along the first row, with signs + − +.

مثال حل‌شده

Medium: cofactor expansion

Find the determinant of [123014560]\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}.

نمایش راه‌حل
  1. Minor of 1 (cover its row and column):
    1⋅0−4⋅6=−241 \cdot 0 - 4 \cdot 6 = -24
  2. Minor of 2:
    0⋅0−4⋅5=−200 \cdot 0 - 4 \cdot 5 = -20
  3. Minor of 3:
    0⋅6−1⋅5=−50 \cdot 6 - 1 \cdot 5 = -5
  4. Combine with signs + − +.
    1(−24)−2(−20)+3(−5)=−24+40−15=11(-24) - 2(-20) + 3(-5) = -24 + 40 - 15 = 1

پاسخ: 11

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مثال حل‌شده

Medium: a zero determinant

Find the determinant of [201132111]\begin{bmatrix} 2 & 0 & 1 \\ 1 & 3 & 2 \\ 1 & 1 & 1 \end{bmatrix}.

نمایش راه‌حل
  1. Expand along the first row; the 0 makes the middle term vanish.
    2(3⋅1−2⋅1)−0+1(1⋅1−3⋅1)2(3 \cdot 1 - 2 \cdot 1) - 0 + 1(1 \cdot 1 - 3 \cdot 1)
  2. Work it out.
    2(1)+1(−2)=02(1) + 1(-2) = 0
  3. The determinant is 0, so this matrix is singular: it has no inverse.

پاسخ: 00 — the matrix is singular

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A shortcut: triangular matrices

If every entry below (or above) the main diagonal is 0, the determinant is just the product of the diagonal. Row reduction uses this: turn the matrix into a triangular one with row operations, keeping track of how each operation changes the determinant (swapping two rows changes the sign; adding a multiple of one row to another changes nothing).

مثال حل‌شده

Hard: read it off the diagonal

Find the determinant of [257031004]\begin{bmatrix} 2 & 5 & 7 \\ 0 & 3 & 1 \\ 0 & 0 & 4 \end{bmatrix}.

نمایش راه‌حل
  1. The matrix is upper triangular, so multiply the diagonal.
    2×3×4=242 \times 3 \times 4 = 24

پاسخ: 2424

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Common mistakes

  • Writing bc−adbc - ad instead of ad−bcad - bc.
  • Forgetting the minus sign on the middle term of a 3×3 expansion.
  • Covering the wrong row and column when finding a minor.
  • Trying to find the determinant of a non-square matrix — only square matrices have one.

پرسش‌های متداول

How do you find the determinant of a matrix?

For 2×2 use ad−bcad - bc. For 3×3 expand along a row: multiply each entry by the determinant of the 2×2 matrix left when you cover its row and column, and combine with signs + − +.

What does a determinant of 0 mean?

The matrix is singular: it has no inverse, its rows are linearly dependent, and a system of equations with it as the coefficient matrix has no solution or infinitely many.

Can I find the determinant of a 4×4 matrix?

Yes, with the same expansion — each term needs a 3×3 determinant — or by row reduction. The determinant calculator does 4×4 too.

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