Determinants

According to Cramer's Rule, the system of equations a1x + b1y = c1, a2x + b2y = c2 can be solved only if

1. The coefficient matrix is singular
2. The determinant of the coefficient matrix is non-zero
3. Both equations are identical
4. The determinant is equal to 1

A system of linear equations has a unique solution if the determinant of its coefficient matrix is

1. Zero
2. Positive only
3. Negative only
4. Non-zero

For any square matrix A, the value of |A times A-transpose| is equal to

1. |A|
2. |A| squared
3. 2|A|
4. 0

If one row of a determinant is entirely zero, then the value of the determinant is

1. 1
2. Equal to the sum of the remaining rows
3. 0
4. Cannot be determined

If A is a square matrix and |A| != 0, then the inverse of A is given by

1. A-inverse = adj(A)
2. A-inverse = (1/|A|) x adj(A)
3. A-inverse = |A| x adj(A)
4. A-inverse = adj(A)/2

If every element of a determinant of order 3 is multiplied by 2, then the new determinant is

1. 2 times the original determinant
2. 4 times the original determinant
3. 6 times the original determinant
4. 8 times the original determinant

If A is a square matrix of order 3 and |A| = 5, then the value of |A-transpose| is

1. -5
2. 0
3. 5
4. 25

The cofactor of the element in the first row and first column of the determinant |2 1 / 3 4| is

1. 2
2. 3
3. 4
4. -4

If one row of a determinant is multiplied by a non-zero constant k, then the value of the determinant becomes

1. Unchanged
2. k times the original determinant
3. k squared times the original determinant
4. Zero

Evaluate the determinant |1 2 3 / 0 1 4 / 5 6 0|

1. 1
2. -1
3. 5
4. 10

If A is a square matrix and |A| = 0, then A is

1. Identity matrix
2. Scalar matrix
3. Non-singular matrix
4. Singular matrix

If any two rows of a determinant are identical, then its value is

1. 1
2. -1
3. 0
4. Cannot be determined

If two rows of a determinant are interchanged, then the value of the determinant becomes

1. Double
2. Half
3. Zero
4. Negative of the original determinant

If |1 2 / 3 x| = -2, then the value of x is

1. 2
2. 4
3. 6
4. 8

The value of the determinant |2 3 / 4 5| is

1. -2
2. 2
3. 10
4. 22

If the vectors a = (1, k, 3), b = (3, –1, –1) and c = (–2, 7, λ) are coplanar, then the value of λ is

1. 0
2. 1
3. –1
4. 2

If α, β, γ are the roots of the equation x³ – 7x + 7 = 0, then the determinant [see original] is equal to

1. 1
2. 0
3. 2
4. none of these

If the system of equations x + y + z = 6, x + 2y + 5z = 9, x + 5y + λz = μ is inconsistent, then the value of p (where λ = p) is

1. –2
2. 0
3. 1/2
4. 2

If the system of equations x + λy = 0, y + λz = 0 and λx + z = 0 where λ ∈ R has a non trivial solution, then the value of λ²+λ is equal to

1. –1
2. 0
3. 1
4. none of these

If the system of equations x + y – 3z = 0, (1+k)x + (2+k)y – 8z = 0, x – (1+k)y + (2+k)z = 0 is inconsistent, then the value of p is

1. –2
2. 0
3. 1/2
4. 2

The lines (x–1)/2 = (y–2)/3 = (z–3)/4 and (x–1)/5 = (y–2)/1 = (z–3)/1 are coplanar if

1. k = 1 or –1
2. k = 0 or –3
3. k = 3 or –3
4. k = 0 or –1

The system of equations αx + y + z = 1, x + αy + z = 1, x + y + αz = 1 will have no solution if α =

1. –2
2. –1
3. 3
4. none of these

If a, b, c are different, one value of x which satisfies [determinant = 0 - see original] is given by

1. x = a
2. x = b
3. x = c
4. x = 0

If α, β, γ are the roots of the equation x³ – 7x + 7 = 0, then [determinant - see original] is equal to

1. 1
2. 0
3. 2
4. none of these

If a, b are roots of x² – px + q = 0 and c, d are roots of x² – rx + s = 0, then roots of [determinant = 0 - see original] are

1. p, r
2. q, s
3. p+q, r+s
4. none of these

If α, β, γ are the roots of x³ + ax² + b = 0, then the value of [specific 3×3 determinant - see original] is

1. a³
2. a³ – 3b
3. a³ + 3b
4. a² – 3b

If α, β, γ are the roots of x³ + ax² + b = 0, then the value of the determinant [See original for the 3×3 determinant] is

1. a³
2. a³ – 3b
3. a³ + 3b
4. a² – 3b

If α, β, γ are the roots of x³ + ax² + b = 0, then the value of [3×3 determinant with α,β,γ - see original] is

1. a³
2. a³ – 3b
3. a³ + 3b
4. a² – 3b

The value of λ for which the system of equations 2x – y – z = 12, x – 2y + z = –4, x + y + λz = 4 has no solution is

1. 3
2. –3
3. 2
4. –2

If the system of equations x + ay = 0, az + y = 0, ax + z = 0 has a non-trivial solution, then

1. a = 1/∛2
2. a = 0
3. a = 1
4. a = –1

The values of λ and μ for which the system of equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ have infinite number of solutions are

1. λ = 3, μ = 10
2. λ = 3, μ ≠ 10
3. λ ≠ 3, μ = 10
4. λ ≠ 3, μ ≠ 10

The points (a, b+c), (b, c+a), (c, a+b) are collinear for

1. all values of a, b, c
2. a + b + c = 0 only
3. a = b = c only
4. none of these

The system of equations x + y + z = 6, x + 2y + 3z = 10 and x + 2y + λz = μ has no solution, if λ is

1. 1
2. –2
3. either –2 or 1
4. none of these