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Which of the following is correct?
The determinant is the scalar computed from the elements of a square matrix. It supplies the crucial property of a matrix: it states whether a given matrix is invertible or not (in this case, nonzero determinant) and singular or degenerate (if it equals to zero). It is defined only for square matricRead more
The determinant is the scalar computed from the elements of a square matrix. It supplies the crucial property of a matrix: it states whether a given matrix is invertible or not (in this case, nonzero determinant) and singular or degenerate (if it equals to zero). It is defined only for square matrices (the number of rows and columns in these matrices is the same).
Determinant is a number attached to a square matrix.
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Given that A = [aᵢⱼ] is a square matrix of order 3 x 3 and |A| = -7, then the value of ∑³₁ ₌ ᵢ aᵢ₂ A ᵢ₂, where Aᵢⱼdemotes the cofactor of element a ᵢⱼis:
We are given that A is a square matrix of order 3x3 and |A| = -7. We are asked to find the value of the sum: ∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂. This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansioRead more
We are given that A is a square matrix of order 3×3 and |A| = -7. We are asked to find the value of the sum:
∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂.
This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansion formula for the determinant of a matrix A is:
|A| = ∑ᵢ aᵢⱼ Aᵢⱼ
where aᵢⱼ is the element of the matrix and Aᵢⱼ is its cofactor. Here, the expansion is along the second column. Thus, the sum becomes,
∑₁³ aᵢ₂ Aᵢ₂ = |A|
As we know that |A| = -7, therefore the value of the sum is -7.
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|A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is
To solve this, let's break it down step by step. We are given the equation |A| = |kA|, where A is a square matrix of order 2. That means it is a 2x2 matrix. We are trying to find the sum of all possible values of k. Key Concepts: 1. Determinant of a matrix: For a square matrix A, the determinant isRead more
To solve this, let’s break it down step by step.
We are given the equation |A| = |kA|, where A is a square matrix of order 2. That means it is a 2×2 matrix. We are trying to find the sum of all possible values of k.
Key Concepts:
1. Determinant of a matrix: For a square matrix A, the determinant is denoted as |A|.
2. Scalar multiplication and determinant:** Let A be any square matrix of order n and k be any scalar, then |kA| = k^n |A|.
Here:
– A is of the order of 2. Thus, n = 2.
– The determinant |kA| is defined as:
|kA| = k² |A|
Now, from |A|=|kA| we have:
|A| = k² |A|
If |A| ≠ 0, then divide both sides by |A|. Then
1 = k²
Hence, k = ±1
If |A| = 0, the equation holds true for all value of k.
Therefore, sum of all the possible values of k
= 1 + (-1)
= 0
Hence, sum of all possible values of k is zero.
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The area of the region bounded by the lines y = mx, x = 1, x = 2 and x-axis is 6 sq. units, then m is equal to
The given region is bounded by the lines y = mx, x = 1, x = 2, and the x-axis. The area enclosed by these boundaries is 6 square units. The definite integral of mx from x = 1 to x = 2 gives us the area under the line y = mx, which is then used to calculate the value of m. We solve the integral. We gRead more
The given region is bounded by the lines y = mx, x = 1, x = 2, and the x-axis. The area enclosed by these boundaries is 6 square units.
The definite integral of mx from x = 1 to x = 2 gives us the area under the line y = mx, which is then used to calculate the value of m.
We solve the integral. We get the area as (3m/2). Therefore, we equate it to 6 and solve for m.
Solving
We multiply both sides by 2 to get 3m=12.
Divide it by 3, and thus we get m = 4.
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If d/dx [f(x)] = ax + b and f(0) = 0, then f(x) is equal to
We are given that d/dx [f(x)] = ax + b Step 1: Integrate both sides To find f(x), integrate the given derivative: f(x) = ∫ (ax + b) dx Using standard integration rules: ∫ ax dx = (a x²)/2 and ∫ b dx = bx Thus, f(x) = (a x²)/2 + bx + C where C is the constant of integration. Step 2: Use the given conRead more
We are given that
d/dx [f(x)] = ax + b
Step 1: Integrate both sides
To find f(x), integrate the given derivative:
f(x) = ∫ (ax + b) dx
Using standard integration rules:
∫ ax dx = (a x²)/2 and ∫ b dx = bx
Thus,
f(x) = (a x²)/2 + bx + C
where C is the constant of integration.
Step 2: Use the given condition f(0) = 0
Substituting x = 0 in the equation:
0 = (a(0)²)/2 + b(0) + C
0 = C
Thus, C = 0, so the final function is:
f(x) = (a x²)/2 + bx
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