?(Calculus required) If the entries of the matrix\(C=\left[\begin{array}{cccc} c_{11}(x)

Chapter 1, Problem 22

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(Calculus required) If the entries of the matrix

\(C=\left[\begin{array}{cccc} c_{11}(x) & c_{12}(x) & \cdots & c_{1 n}(x) \\ c_{21}(x) & c_{22}(x) & \cdots & c_{2 n}(x) \\ \vdots & \vdots & & \vdots \\ c_{m 1}(x) & c_{m 2}(x) & \cdots & c_{m n}(x) \end{array}\right] \)

are differentiable functions of x, then we define

\(\frac{d C}{d x}=\left|\begin{array}{cccc} c_{11}^{\prime}(x) & c_{12}^{\prime}(x) & \cdots & c_{1 n}^{\prime}(x) \\ c_{21}^{\prime}(x) & c_{22}^{\prime}(x) & \cdots & c_{2 n}^{\prime}(x) \\ \vdots & \vdots & & \vdots \\ c_{m 1}^{\prime}(x) & c_{m 2}^{\prime}(x) & \cdots & c_{m n}^{\prime}(x) \end{array}\right| \)

Show that if the entries in 𝐴 and 𝐵 are differentiable functions of x and the sizes of the matrices are such that the stated operations can be performed, the

a. \(\frac{d}{d x}(k A)=k \frac{d A}{d x}\)

b. \(\frac{d}{d x}(A+B)=\frac{d A}{d x}+\frac{d B}{d x}\)

c. \(\frac{d}{d x}(A B)=\frac{d A}{d x} B+A \frac{d B}{d x}\)

Equation Transcription:

Text Transcription:

C=[C_11(X) C_12(X) . . .  C_1n(X) C_21(X) C_22(X) . . .  C_2n(X)

C_m1(X) C_m2(X) . . .  C_mn(X)]

dC/dx=[C'_11(X) C'_12(X) . . .  C'_1n(X) C'_21(X) C'_22(X) . . .  C'_2n(X)

C'_m1(X) C'_m2(X) . . .  C'_mn(X)]

d/dx(kA)=kdA/dx

d/dx(A+B)=dA/dx+dB/dx

d/dx(AB)=dA/dxB+AdB/dx