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Wize University Linear Algebra Textbook > Determinants

Determinants and EROs

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Determinant & EROs

Swapping Two Rows

If A→ swap two rows BA \xrightarrow{\text{ swap two rows }} B, then det(B)=−det(A)\det{B} = -\det{A},
Example
det[1234]=−det[3412]{\rm det} \begin{bmatrix} 1&2\\3&4 \end{bmatrix} = -{\rm det} \begin{bmatrix} 3&4\\1&2 \end{bmatrix}
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Multiplying a Row by a Non-Zero Constant

If A→ multiply a row by k BA \xrightarrow{\text{ multiply a row by } k\ } B, then det(B)=k⋅det(A)\det{B} = k \cdot \det{A}.
Example
det[1268]=2⋅det[1234]{\rm det} \begin{bmatrix} 1&2\\6&8 \end{bmatrix} = 2 \cdot {\rm det} \begin{bmatrix} 1&2\\3&4 \end{bmatrix}
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Adding a Multiple of One Row to Another Row

If A→ add a multiple of one row to another row BA \xrightarrow{\text{ add a multiple of one row to another row }} B, then det(B)=det(A)\det{B}=\det{A}
Example
det[1100]=det[11−1−1]{\rm det} \begin{bmatrix} 1&1\\0&0 \end{bmatrix} = {\rm det} \begin{bmatrix} 1&1\\-1&-1 \end{bmatrix}

Wize Tip
These determinant properties hold even if the operations are performed on the columns instead of the rows.

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Example: Determinants & EROs

The following EROs were performed on AA to get to BB:
A→−3R2A1→R3: R3+R1A2→R1↔R2BA\xrightarrow{-3R_2}A_1\xrightarrow{R_3:\ R_3+R_1}A_2\xrightarrow{R_1\leftrightarrow R_2}B
If det(A)=5\det A = 5, find det(B)\det B.
Let's work backward, starting from BB.
Swapping rows means the determinant is negated (multiply by -1): det(B)=−det(A2)\det B =-\det{A_2}
Next, we added a multiple of one row to another which does not change the determinant: det(A2)=det(A1)\det{A_2} = \det{A_1}
Then we multiplied a row by -3, so the determinant is multiplied by -3: det(A1)=(−3)⋅det(A)\det{A_1} = (-3)\cdot\det A
Putting it all together:

det(B)=−det(A2)⏟swapped rows=−det(A1)⏟added multiple of row=−(−3)⋅det(A)⏟multiplied row by −3=−(−3)(5)=15\det B \quad = \quad \underbrace{-\det{A_2}}_\text{swapped rows} \quad= -\underbrace{\det{A_1}}_\text{added multiple of row} \quad= -\underbrace{(-3)\cdot \det{A}}_{\text{multiplied row by }-3} \quad= -(-3)(5)=\boxed{15}
A=[abcpqrxyz],B=[4x4y4zp+xq+yr+zabc]A= \begin{bmatrix} a&b&c\\ p&q&r\\ x&y&z \end{bmatrix}, \quad B= \begin{bmatrix} 4x&4y&4z\\ p+x&q+y&r+z\\ a&b&c \end{bmatrix}

If det(A)=3\det A=3, find det(B)\det B.
Extra Practice
133 - FML 3 - 18.1W e.g. 17.1
If   ⁣A‾\bcb{\boldsymbol{ \ul{A} }} is a 3×3\bcb{\boldsymbol{ 3 \times 3}} matrix with det(  ⁣A‾)=−4\bcb{\boldsymbol{ \det{\ul{A}} = -4}}, find:
det(2  ⁣A‾)\boldsymbol{ \det {2\ul{A}}}
133 - FML 3 - 18.1W e.g. 9.3
Given that ∣abcdefghi∣=−3\bcb{\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = -3}, find
∣  abc2d2e2fdef∣{ \begin{vmatrix}  a & b & c \\ 2d & 2e & 2f \\ d & e & f \end{vmatrix} }
133 - FML 3 - 18.1W e.g. 9.2
Given that ∣abcdefghi∣=−3\bcb{\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = -3}, find
∣3a3b3cg+3ah+3b  i+3c2d2e2f∣{ \begin{vmatrix} 3a & 3b & 3c \\ g + 3a & h+3b  & i + 3c \\ 2d & 2e & 2f \end{vmatrix} }
133 - FML 3 - 18.1W e.g. 9.1
Given that ∣abcdefghi∣=−3\bcb{\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = -3}, find
∣−ad2g−be2h−cf2i∣{ \begin{vmatrix} - a & d & 2 g \\ - b & e & 2h \\ -c & f & 2i \end{vmatrix} }
Determinants and EROs
A=[abcpqrxyz],B=[4x4y4zp+xq+yr+zabc]A= \begin{bmatrix} a&b&c\\ p&q&r\\ x&y&z \end{bmatrix}, \quad B= \begin{bmatrix} 4x&4y&4z\\ p+x&q+y&r+z\\ a&b&c \end{bmatrix}
If det(A)=3\det A=3, find det(B)\det B.
Determinants and Elementary Row Operations (EROs)
Calculate determinant of [−200000081492−1000103041000]\begin{bmatrix} -2&0&0&0&0\\ 0&0&8&1&4\\ 9&2&-1&0&0\\ 0&1&0&3&0\\ 4&1&0&0&0\\ \end{bmatrix}.
Determinants and Elementary Row Operations (EROs)
If A=[abcdefghi]A=\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} and det A=−2det\ A=-2, find det[ghi2d−a2e−b2f−cabc]det \begin{bmatrix} g&h&i\\ 2d-a&2e-b&2f-c\\ a&b&c \end{bmatrix}.
(Duplicated)
If it is known that A=[abcpqrxyz]A= \begin{bmatrix} a&b&c\\ p&q&r\\ x&y&z \end{bmatrix} and det(A)=5det(A)=5, determine det[ac2bpr2q2x2z4y]det \begin{bmatrix} a&c&2b\\ p&r&2q\\ 2x&2z&4y\\ \end{bmatrix}.
Determinants and Elementary Row Operations (EROs)
If it is known that A=[abcpqrxyz]A= \begin{bmatrix} a&b&c\\ p&q&r\\ x&y&z \end{bmatrix} and det(A)=5det(A)=5, determine det[2a2b2cp−3xq−3yr−3zpqr]det \begin{bmatrix} 2a&2b&2c\\ p-3x&q-3y&r-3z\\ p&q&r \end{bmatrix}.