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) = a0 + a1x + a2x2+…+ an−1xn−1 + anxn, then the determinant of A is
matrix ‘A’ ⇒ f(x) = a0 + a1x + a2x2 +……….+an−1xn−1 + anxnf(x)
if 2 × 2 matrix then characteristic equation ⇒ aλ2 + bλ + c = 0
⇒ |A| = λ1.λ2. = c/a
where ao = c;an = a2 = a
if 3 × 3 matrix → then characteristic equation ⇒ aλ3 + bλ2 + cλ + d = 0
∴ For n × n matrix, |A| = λ1.λ2.λ3….λn
What is the determinant of the below-given 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:
|A||A| 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 A3 − 9A2 − 47A?
Characteristic equation of the given matrix is
Every matrix satisfies its own characteristic equation
∴A3 − 9A2 − 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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