Theory of Intelligence
11TH•Mathematics•Matrices And Determinants•Exercise 4.2

4.2 Matrix Operations

Practice set 4.2 (determinants: cofactors, properties, inverse, and transpose).

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Q4

Question 4

If A=[12−30−50−2−27]A=\begin{bmatrix}1&2&-3\\0&-5&0\\-2&-2&7\end{bmatrix} and B=[−5−25−3−14−2−12]B=\begin{bmatrix}-5&-2&5\\-3&-1&4\\-2&-1&2\end{bmatrix}, then find:

Select Part:Part (i) of 2
(i)

A31, A32, A33A_{31},\,A_{32},\,A_{33} and ∣A∣|A|

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Step-by-Step Solution

Question 4 • Part (i)

verifiedVerified Answer

Part (i): Cofactors of Matrix AA & Row 3 Determinant Expansion

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Cofactor & Determinant Lab • Exercise 4.2 Q4

Cofactor Expansion of Matrix A (Row 3)

Explore how minors M_{3j} and cofactors A_{3j} combine along row 3 to compute the exact determinant volume.

Select Problem Matrix:
Inspect Element:
Step 1: Row 3 & Column 1 Cross-OutElement (3, 1) = -2
Minor 2×2 SubmatrixCrossed-OutTarget a₃ⱼ
Step 2: Minor M₃1 & Cofactor A₃1Sign (-1)³⁺1 = +1
Minor Submatrix M₃1
2-3-50
⇒
Formula: ad - bc
(2)(0) - (-3)(-5)
= 0 - 15 = -15
Cofactor A₃1 = (-1)³⁺1 · M₃1-15
Contribution to |A| = a₃1 · A₃1 = (-2) · (-15) = 30
Row 3 Expansion: |A| = a₃₁A₃₁ + a₃₂A₃₂ + a₃₃A₃₃|A| = -5
Term 1: a₃1 · A₃1
(-2) · (-15) = 30
Term 2: a₃2 · A₃2
(-2) · (0) = 0
Term 3: a₃3 · A₃3
(7) · (-5) = -35
Sum: 30 + (0) + (-35) = -5Matches textbook verified answer |A| = -5

Step-by-Step Solution

Given matrix AA:

A=[12−30−50−2−27]A = \begin{bmatrix} 1 & 2 & -3 \\ 0 & -5 & 0 \\ -2 & -2 & 7 \end{bmatrix}

The cofactor of element aija_{ij} is defined as Aij=(−1)i+jMijA_{ij} = (-1)^{i+j} M_{ij}, where MijM_{ij} is the minor determinant obtained by deleting row ii and column jj.

1. Compute Minor and Cofactor A31A_{31}

Delete Row 3 and Column 1:

M31=∣2−3−50∣=(2)(0)−(−3)(−5)=0−15=−15M_{31} = \begin{vmatrix} 2 & -3 \\ -5 & 0 \end{vmatrix} = (2)(0) - (-3)(-5) = 0 - 15 = -15 A31=(−1)3+1M31=(1)(−15)=−15A_{31} = (-1)^{3+1} M_{31} = (1)(-15) = -15

2. Compute Minor and Cofactor A32A_{32}

Delete Row 3 and Column 2:

M32=∣1−300∣=(1)(0)−(−3)(0)=0M_{32} = \begin{vmatrix} 1 & -3 \\ 0 & 0 \end{vmatrix} = (1)(0) - (-3)(0) = 0 A32=(−1)3+2M32=(−1)(0)=0A_{32} = (-1)^{3+2} M_{32} = (-1)(0) = 0

3. Compute Minor and Cofactor A33A_{33}

Delete Row 3 and Column 3:

M33=∣120−5∣=(1)(−5)−(2)(0)=−5M_{33} = \begin{vmatrix} 1 & 2 \\ 0 & -5 \end{vmatrix} = (1)(-5) - (2)(0) = -5 A33=(−1)3+3M33=(1)(−5)=−5A_{33} = (-1)^{3+3} M_{33} = (1)(-5) = -5

4. Expand Determinant ∣A∣|A| Along Row 3

∣A∣=a31A31+a32A32+a33A33=(−2)(−15)+(−2)(0)+(7)(−5)=30+0−35=−5\begin{aligned} |A| &= a_{31} A_{31} + a_{32} A_{32} + a_{33} A_{33} \\ &= (-2)(-15) + (-2)(0) + (7)(-5) \\ &= 30 + 0 - 35 = -5 \end{aligned} A31=−15,A32=0,A33=−5,∣A∣=−5\boxed{A_{31} = -15, \quad A_{32} = 0, \quad A_{33} = -5, \quad |A| = -5}
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