$$For1$$≤i, j≤$$3$$,$$Letaij=$$∫−π$$/2$$π$$/2cos(ix)$$$$cos$$$$(jx)dxand$$ let $$A=$$[aij]$$3$$×$$3then$$

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$$For1$$≤i, j≤$$3$$,$$Letaij=$$∫−π$$/2$$π$$/2cos(ix)$$$$cos$$$$(jx)dxand$$ let $$A=$$[aij]$$3$$×$$3then$$

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For $$1$$≤i≤$$3$$,$$aii=$$∫$$-$$π$$/2$$π$$/2cos2(ix)dx=2$$∫$$0$$π$$/2cos2(ix)dx=$$∫$$0$$π$$/2$$[$$1+$$$$cos$$(2ix)]$$dx=x+$$$$sin$$(2ix)2i0$$π$$/2=$$π$$2+0=$$π$$2$$ For $$1$$≤i,j≤$$3$$,i≠$$jaij=2$$∫$$0$$π$$/2cos(ix)$$cos$$(jx)dx=$$∫$$0$$π$$/2{$$$$cos$$[$$(i+j)x$$]$$+$$$$cos$$[$$(i-j)x$$]$$}dx=$$$$sin$$[$$(i+j)x$$]$$i+j+$$$$sin$$[$$(i-j)x$$]$$i-j0$$π$$/2=$$$$sin$$[$$(i+j)π$$/2$$]$$i+j+$$$$sin$$[(i-j)π$$/2$$]$$i-j$$∴$$a12=-13+1=23=a21a13=0=a31a23=15+1=65=a32$$ Thus, $$A=$$π$$/22/302/3$$π$$/26/506/5$$π$$/2$$|A|$$=$$π$$28-43$$$$π$$$$72$$≠$$0$$,$$A-1$$ exists. ∴ $$AX=Bhas$$ a unique solution, for every matrix B

For $1 \leq i, j \leq 3$ ,Let $a_{i j} = \int_{- π / 2}^{π / 2} \cos (i x) \cos (j x) d x$ and let $A = {[a_{i j}]}_{3 \times 3}$ then

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For1≤i, j≤3,Letaij=∫−π/2π/2cos(ix)cos(jx)dxand let A=[aij]3×3then

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For $1 \leq i, j \leq 3$ ,Let $a_{i j} = \int_{- π / 2}^{π / 2} \cos (i x) \cos (j x) d x$ and let $A = {[a_{i j}]}_{3 \times 3}$ then