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If A is a square matrix and A² = A, then (I + A)² – 3A is equal to:
Given that A is a square matrix and A² = A, we must reduce (I + A)² - 3A. Step 1: Expand (I + A)² (I + A)² = I² + 2IA + A² Given that I² = I and A² = A, we get: (I + A)² = I + 2A + A (I + A)² = I + 3A Step 2: Subtract 3A Substitute this back into the expression: (I + A)² - 3A = (I + 3A) - 3A (I + A)Read more
Given that A is a square matrix and A² = A, we must reduce (I + A)² – 3A.
Step 1: Expand (I + A)²
(I + A)² = I² + 2IA + A²
Given that I² = I and A² = A, we get:
(I + A)² = I + 2A + A
(I + A)² = I + 3A
Step 2: Subtract 3A
Substitute this back into the expression:
(I + A)² – 3A = (I + 3A) – 3A
(I + A)² – 3A = I
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Given that matrices A and B are of order 3 x n and m x 5 respectively, then the order of matrix C = 5A + 3B is:
Matrix A has order 3 × n, and B has order m × 5; find the order of matrix C = 5A + 3B Step 1: Rules governing addition of matrices Addition is defined only when both matrices are the same order, and the same with our scenario of A versus B. Matrix A has order 3 × n and matrix B has order m × 5. ForRead more
Matrix A has order 3 × n, and B has order m × 5; find the order of matrix C = 5A + 3B
Step 1: Rules governing addition of matrices
Addition is defined only when both matrices are the same order, and the same with our scenario of A versus B.
Matrix A has order 3 × n and matrix B has order m × 5.
For the addition 5A + 3B to be possible, we must have n = m, meaning both matrices must have the same number of columns.
Step 2: Order of matrix C
Once the condition n = m is met, then the matrix C that results from it will be of the same order as that of A and B, which is 3 × 5.
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If order of matrix A is 2 x 3 of matrix B is 3 x 2, and of matrix C is 3 x 3, then which one of the following is not defined?
- Matrix A has order 2 × 3 and matrix B has order 3 × 2. - The transpose of matrix B, B', has order 2 × 3. - For the sum A + B', both matrices have order 2 × 3, so the addition is possible. - However, matrix C has order 3 × 3, and for matrix multiplication to be defined, the number of columns in matRead more
– Matrix A has order 2 × 3 and matrix B has order 3 × 2.
– The transpose of matrix B, B’, has order 2 × 3.
– For the sum A + B’, both matrices have order 2 × 3, so the addition is possible.
– However, matrix C has order 3 × 3, and for matrix multiplication to be defined, the number of columns in matrix C (3) must match the number of rows in A + B’ (2).
– Since 3 ≠ 2, multiplying matrix C by A + B’ is not defined.
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If P is a 3 x 3 matrix such that P’ = 2P + I, where P’ is the transpose of P, then
We are given that P is a 3 × 3 matrix such that P' = 2P + I, where P' is the transpose of P. Step 1: Take the transpose of both sides We take the transpose of both sides of the equation P' = 2P + I: (P')' = (2P + I)' Since the transpose of the transpose of a matrix is the matrix itself, we get: P =Read more
We are given that P is a 3 × 3 matrix such that P’ = 2P + I, where P’ is the transpose of P.
Step 1: Take the transpose of both sides
We take the transpose of both sides of the equation P’ = 2P + I:
(P’)’ = (2P + I)’
Since the transpose of the transpose of a matrix is the matrix itself, we get:
P = 2P’ + I
Step 2: Plug P’ = 2P + I into this equation
Next, plug the expression P’ = 2P + I into the equation:
P = 2(2P + I) + I
P = 4P + 2I + I
P = 4P + 3I
Step 3: Move terms around
Now we move the terms around in the equation.
P – 4P = 3I
-3P = 3I
P = -I
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If y = tan⁻¹ (e²ˣ), then dy/dx is equal to
We are given y = tan⁻¹(e²ˣ) and need to find dy/dx. Step 1: Differentiate both sides with respect to x We differentiate the equation y = tan⁻¹(e²ˣ) using the chain rule. The derivative of tan⁻¹(u) with respect to u is 1/(1 + u²), so: dy/dx = 1 / (1 + (e²ˣ)²) * d/dx(e²ˣ) Step 2: Differentiate e²ˣ TheRead more
We are given y = tan⁻¹(e²ˣ) and need to find dy/dx.
Step 1: Differentiate both sides with respect to x
We differentiate the equation y = tan⁻¹(e²ˣ) using the chain rule. The derivative of tan⁻¹(u) with respect to u is 1/(1 + u²), so:
dy/dx = 1 / (1 + (e²ˣ)²) * d/dx(e²ˣ)
Step 2: Differentiate e²ˣ
The derivative of e²ˣ with respect to x is:
d/dx(e²ˣ) = 2e²ˣ
Step 3: Substitute into the derivative
Substitute this back into the expression for dy/dx:
dy/dx = 1 / (1 + e⁴ˣ) * 2e²ˣ
dy/dx = 2e²ˣ / (1 + e⁴ˣ)
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