Find the value of x for the singular matrix?
Determinant of a singular matrix is 0
∴ 3×15×(24−3x)=0
∴ 3x=24
∴ x=8
If the determinant of the matrix is 26, then the determinant of the matrix is
If two rows are interchanged in a determinant then the value of the determinant does not change but sign will change. In the given question, first and third rows are interchanged.
Determinant wil be 26
The value of the determinant
If a matrix A is given by f(x) = a_{0} + a_{1}x + a_{2}x^{2}+…+ a_{n−1}x^{n−1} + a_{n}x^{n}, then the determinant of A is
matrix ‘A’ ⇒ f(x) = a_{0 }+ a_{1}x + a_{2}x^{2 }+……….+a_{n−1}x^{n−1 }+ a_{n}x^{n}f(x)
if 2 × 2 matrix then characteristic equation ⇒ aλ^{2 }+ bλ + c = 0
⇒ A = λ_{1}.λ_{2}. = c/a
where a_{o} = c;a_{n} = a_{2} = a
if 3 × 3 matrix → then characteristic equation ⇒ aλ^{3 }+ bλ^{2 }+ cλ + d = 0
∴ For n × n matrix, A = λ_{1}.λ_{2}.λ_{3}….λ_{n}
What is the determinant of the belowgiven matrix?
Determinant of the given matrix is
Δ = (a − b) (a − c) (a − d) (b − c) (b − d)(c − d)
a = 3, b = 5, c = 7, d = 9
∴ Δ = 768
If A = then the value of adj (adj (A))?
A is upper triangular matrix
∴ product of diagonal elements = A
= 1 × 5 × 8 × 10 = 400
Note:
AA is determinant of matrix A
What is the rank of A−B where
∴ rank is < 3
Also
Hence rank of is A−B is 2
What is the value of A^{3 }− 9A^{2 }− 47A?
Characteristic equation of the given matrix is
Every matrix satisfies its own characteristic equation
∴A^{3} − 9A^{2} − 47A = 20I
The factorized form of the following determinant is:
Applying R2 → R2 – R1
Now, expanding from a11.
= (m  l)(n  l)(n + l  m  l)
= (m  l)(n  l)(n  m)
A 3 × 5 matrix has all its entries equal to 1. The rank of the matrix is
Rank ≤ 3 since all entries are 1,
∴ 3 × 3 submatrices  =  2 × 2 submatrices  = 0
only 1 × 1 or single entries are ≠ 0
∴ rank = 1
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