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Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#215;</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula> Foeplitz matrix is the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>th Fibonacci number, while the inverse matrix of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#215;</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula> Foeplitz matrix is sparse and can be expressed by the <i>n</i>th and the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>th Fibonacci number. Similarly, the determinant of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#215;</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula> Loeplitz matrix can be expressed by use of the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>th Lucas number, and the inverse matrix of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#215;</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula><inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>&gt;</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results.