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For a 3xx3 matrix A if det A=4, then det...

For a `3xx3` matrix A if det A=4, then det (Adj. A) equals

A

`-4`

B

`4`

C

`16`

D

`64`

Text Solution

AI Generated Solution

The correct Answer is:
To find the determinant of the adjoint of a \(3 \times 3\) matrix \(A\) given that \(\text{det}(A) = 4\), we can use the following formula: \[ \text{det}(\text{Adj} A) = (\text{det} A)^{n-1} \] where \(n\) is the order of the matrix. In this case, since \(A\) is a \(3 \times 3\) matrix, \(n = 3\). ### Step-by-Step Solution: 1. **Identify the order of the matrix**: \[ n = 3 \] 2. **Use the formula for the determinant of the adjoint**: \[ \text{det}(\text{Adj} A) = (\text{det} A)^{n-1} \] 3. **Substitute the given value of \(\text{det}(A)\)**: \[ \text{det}(\text{Adj} A) = (4)^{3-1} \] 4. **Calculate \(3 - 1\)**: \[ 3 - 1 = 2 \] 5. **Now calculate \(4^2\)**: \[ 4^2 = 16 \] 6. **Conclusion**: \[ \text{det}(\text{Adj} A) = 16 \] ### Final Answer: \[ \text{det}(\text{Adj} A) = 16 \]
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Knowledge Check

  • If A is a 3 xx 3 matrix such that |A| = 4, then what is A(adj A) equal to ?

    A
    `[{:(1,0,0),(0,1,0),(0,0,1):}]`
    B
    `[{:(4,0,0),(0,4,0),(0,0,4):}]`
    C
    `[{:(16,0,0),(0,16,0),(0,0,16):}]`
    D
    Cannot be determined, as data is insufficient.

  • if A is a square matrix such that A^(2)=A, then det (A) is equal to

    A
    0 or 1
    B
    `-2 or 2`
    C
    `-3 or 3`
    D
    none of these

  • If A is a 3xx3 matrix such that det.A=0, then

    A
    A=0
    B
    A is non-singular
    C
    all elements of A are equal
    D
    A is singular

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