04 ਨਿਰਧਾਰਕ
ਫੁਟਕਲ ਅਭਿਆਸ
Question:
Without expanding the determinant, prove that |aa2bcbb2cacc2ab| =|1a2a31b2b31c2c3|
Answer:
Solution:
We can use the Laplace Expansion Theorem to prove this statement. According to the theorem, the determinant of a matrix is equal to the sum of the products of the elements of any row or column and its corresponding cofactor.
Therefore, we can expand the determinant using the first column of the given matrix and its corresponding cofactor.
|aa2bcbb2cacc2ab|
= a|1a2bcbb2cacc2ab|
- b|aa21bb2cacc2ab|
- c|aa2bcbb21cc2ab|
= aUnexpected text node: '<mn'
Question:
Let A=|1-21-231115|Verify that (i)[adj A]−1=adj(A)−1 (ii)(A-1)-1=A
Answer:
Step 1: Calculate the adjugate of A
adj A = |32-12-1-1-1-11|
Step 2: Calculate the inverse of A
A-1 = |32-12-1-1-1-11|
Step 3: Verify that (i) adj A-1 = adj (A-1)
adj A-1 = |32-12-1-1-1-11|
adj (A-1) = |32-12-1-1-1-11|
Since adj A-1 = adj (A-1), the first part of the statement is true.
Step 4: Verify that (ii) (A-1)-1 = A
(A-1)-1 = Unexpected text node: '<mtr'
Question:
Using properties of determinants, prove that: |αα2β+γββ2γ+αγγ2α+β| =(α−β)(β−γ)(γ−α)(α+β+γ)
Answer:
- Expand the determinant using the Laplace Expansion along the last column:
|αα2β+γββ2γ+αγγ2α+β|=(α+β+γ)|αα2βββ2γγγ2α|
- Apply the rule of Sarrus:
|αα2βββ2γγγ2α|=αα2+ββ2+γγ2+2αβγ-αβ2-βγ2-γα2
- Apply the distributive property:
(α+β+γ)(α2+β2+</Unknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: h2Unknown node type: olUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: olUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: pUnknown node type: h2Unknown node type: pUnknown node type: h2Unknown node type: ol